(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 9.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 358392, 9538] NotebookOptionsPosition[ 339746, 9214] NotebookOutlinePosition[ 340355, 9237] CellTagsIndexPosition[ 340280, 9232] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "February 24, 25 and 27, 2025\nEnumerative combinatorics and effective \ aspects of differential equations, CIRM, Marseille, France\n", StyleBox["Summation theory of difference rings and applications", FontSize->48], "\nCarsten Schneider\n" }], "Subsubsection", CellChangeTimes->{{3.949316926934705*^9, 3.94931692693572*^9}, { 3.9493170371981373`*^9, 3.949317099753941*^9}, {3.949317168364214*^9, 3.949317188141356*^9}}, TextAlignment->Center,ExpressionUUID->"cd936624-eb53-4478-b2c8-5197555d265d"], Cell[BoxData[ RowBox[{"Exit", "[", "]"}]], "Input", CellChangeTimes->{{3.695385043505473*^9, 3.6953850452084837`*^9}}, CellLabel->"In[21]:=",ExpressionUUID->"6184fcd3-f04e-49e5-8294-ce8a0e08da47"], Cell[BoxData[ RowBox[{ RowBox[{"SetDirectory", "[", "\"\<~/Packages\>\"", "]"}], ";"}]], "Input", CellChangeTimes->{{3.6120727995066547`*^9, 3.612072821282559*^9}, { 3.612072857851555*^9, 3.6120728593405943`*^9}, {3.6780887504054127`*^9, 3.6780887538731337`*^9}, 3.678146273301935*^9}, CellLabel->"In[1]:=",ExpressionUUID->"2c075768-366e-4a3e-be1f-e693a1710860"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"<<", "\"\\""}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"<<", "HarmonicSums.m"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"<<", "EvaluateMultiSums.m"}], ";"}]}], "Input", CellChangeTimes->{{3.571998331919552*^9, 3.571998333534644*^9}, { 3.612072616027573*^9, 3.612072650706869*^9}, {3.612072733074481*^9, 3.612072735923788*^9}, 3.612072864805501*^9, {3.695358398582679*^9, 3.6953584184761*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"09e10ab0-75ab-47fa-8de9-135f00ab5c40"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["Sigma", FontColor->RGBColor[1, 0, 0]], RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"-", " ", "A"}], " ", "summation", " ", "package", " ", "by", " ", "Carsten", " ", "Schneider"}], " ", "\[LongDash]", " ", RowBox[{"\[Copyright]", " ", "RISC"}], " ", "\[LongDash]", " ", RowBox[{"V", " ", "2.9", " ", RowBox[{"(", RowBox[{ RowBox[{"June", " ", "5"}], ",", " ", "2024"}], ")"}], " ", "click", " ", "for", " "}]}]}], ButtonBox[ StyleBox["Help", "Hyperlink", FontVariations->{"Underline"->True}], Appearance->{Automatic, None, "Normal", Automatic}, ButtonFunction:>Sigma`Version`Private`SigmaHelp[], Evaluator->Automatic]}]], "Print", CellFrame->0.5, CellChangeTimes->{3.949641091510189*^9}, FontColor->RGBColor[0, 0, 0], Background->RGBColor[0.796887, 0.789075, 0.871107], ButtonBoxOptions->{ Active->True},ExpressionUUID->"ae50fa83-f872-4430-9d83-5b6413a94f12"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"HarmonicSums", " ", "by", " ", "Jakob", " ", "Ablinger", " ", RowBox[{"(", RowBox[{"modified", " ", "by", " ", "CS"}], ")"}]}], " ", "\[LongDash]", " ", RowBox[{"\[Copyright]", " ", "RISC"}], " ", "\[LongDash]", " ", RowBox[{"Version", " ", "1.01"}]}], RowBox[{"(", RowBox[{ RowBox[{"05", "/", "07"}], "/", "23"}], ")"}]}], ButtonBox[ StyleBox["Help", "Hyperlink", FontVariations->{"Underline"->True}], Appearance->{Automatic, None, "Normal", Automatic}, ButtonFunction:>HarmonicSums`Private`HarmonicSumsHelp[], Evaluator->Automatic]}]], "Print", CellFrame->0.5, CellChangeTimes->{3.949641093936754*^9}, FontColor->GrayLevel[0.], Background->RGBColor[0.796887, 0.789075, 0.871107], ButtonBoxOptions->{ Active->True},ExpressionUUID->"20512962-db41-4a5b-941f-e537bb0f5fe2"], Cell[BoxData[ RowBox[{ RowBox[{ "EvaluateMultiSums", " ", "by", " ", "Carsten", " ", "Schneider", " ", RowBox[{"(", RowBox[{ RowBox[{"still", " ", "under", " ", "development"}], "..."}], ")"}]}], " ", "\[LongDash]", " ", RowBox[{"\[Copyright]", " ", "RISC"}], " ", "\[LongDash]", " ", RowBox[{"V", " ", "0.996", " ", RowBox[{"(", RowBox[{ RowBox[{"March", " ", "11"}], ",", " ", "2022"}], ")"}]}]}]], "Print", CellFrame->0.5, CellChangeTimes->{3.949641094969063*^9}, FontColor->RGBColor[0, 0, 0], Background->RGBColor[0.796887, 0.789075, 0.871107], ButtonBoxOptions->{ Active->True},ExpressionUUID->"e9c0ec06-8bb5-49f2-84cf-8dab8d23ed54"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Part 1", "Title", CellChangeTimes->{{3.9493180857775497`*^9, 3.949318087669476*^9}},ExpressionUUID->"e75bf771-6de1-4cf6-88bb-\ 4df1e56cb17e"], Cell[BoxData[" "], "Input", CellChangeTimes->{3.9493208430151663`*^9}, CellLabel->"In[5]:=",ExpressionUUID->"5a582f38-fe00-4067-90e5-9679c084050d"], Cell[CellGroupData[{ Cell["Warm up example", "Section", CellChangeTimes->{{3.560426203265094*^9, 3.560426208408922*^9}}, FontSize->36,ExpressionUUID->"92b73221-6725-4cc5-b951-f214acafd935"], Cell[CellGroupData[{ Cell["The inner sum", "Subsection", CellChangeTimes->{{3.949319458553813*^9, 3.949319459367777*^9}},ExpressionUUID->"dabbcba7-cdd2-460f-ab04-\ 37fdbe1ebfcb"], Cell[BoxData[ RowBox[{ RowBox[{"summand", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", "!"}], " ", RowBox[{"k", "!"}], " ", RowBox[{"n", "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "n"}], ")"}], "!"}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "n"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "n"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "n"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "n"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "n"}], ")"}], "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], "!"}]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.695358996200685*^9, 3.695358999573455*^9}, { 3.949319475159174*^9, 3.949319479591729*^9}, {3.949319543396196*^9, 3.949319569875986*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"49705659-a4e2-4d4d-974e-d352435df9e1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"innerSum", "=", RowBox[{"SigmaSum", "[", RowBox[{"summand", ",", RowBox[{"{", RowBox[{"j", ",", "0", ",", "a"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.9493189948644047`*^9, 3.9493190204949408`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"43cd98a7-3d11-480e-ac6e-8fd966726e7d"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "a"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", "!"}], " ", RowBox[{"k", "!"}], " ", RowBox[{"n", "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "n"}], ")"}], "!"}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "n"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "n"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "n"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "n"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "n"}], ")"}], "!"}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], "!"}]}], ")"}]}]}]], "Output",\ CellChangeTimes->{3.949641101124289*^9}, CellLabel->"Out[7]=",ExpressionUUID->"ea399b29-ee73-448f-ab4d-d42673858680"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"closedForm", "=", RowBox[{"SigmaReduce", "[", "innerSum", "]"}]}]], "Input", CellChangeTimes->{{3.5265777354953413`*^9, 3.526577771624408*^9}, { 3.949319182288776*^9, 3.949319197157683*^9}, 3.949319358477303*^9}, CellLabel->"In[8]:=",ExpressionUUID->"1c6215b7-9764-404a-a742-5945fa1f1c22"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"2", "+", RowBox[{"2", " ", "a"}], "+", "k", "+", "n"}], ")"}], " ", RowBox[{"a", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"a", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "a"}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n", "+", RowBox[{"k", " ", "n"}]}], ")"}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"a", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"a", "+", "k"}]}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n", "+", RowBox[{"k", " ", "n"}]}], ")"}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"a", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"a", "+", "n"}]}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", "k", "-", "n", "-", RowBox[{"k", " ", "n"}]}], ")"}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"k", "+", "n"}]}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"a", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"n", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"a", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"a", "+", "k", "+", "n"}]}], "]"}]}], RowBox[{"k", " ", "n", " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], SubscriptBox["!", "."]}]}]]}]], "Output", CellChangeTimes->{3.949641104605047*^9}, CellLabel->"Out[8]=",ExpressionUUID->"4f8d8377-216e-4776-94dd-9f390d58cfe3"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaLimit", "[", RowBox[{"closedForm", ",", RowBox[{"{", RowBox[{"k", ",", "n"}], "}"}], ",", "a"}], "]"}]], "Input", CellChangeTimes->{{3.949319030938262*^9, 3.949319077262678*^9}, { 3.949319350183763*^9, 3.949319351572194*^9}, {3.949319492749*^9, 3.949319493513942*^9}}, CellLabel->"In[9]:=",ExpressionUUID->"53d59f9a-ad82-48f7-b23b-d0644617fb96"], Cell[BoxData["\<\"Loading Relations...\"\>"], "Print", CellChangeTimes->{3.9496411050295563`*^9}, CellLabel-> "During evaluation of \ In[9]:=",ExpressionUUID->"104806a6-0ca9-4d41-a3aa-d868b92e38d7"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]], "+", FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]], "-", FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"k", "+", "n"}]}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]]}]], "Output", CellChangeTimes->{3.9496411089869423`*^9}, CellLabel->"Out[9]=",ExpressionUUID->"fe340302-4735-485e-8690-fa0367f50df3"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GetFullDefinition", "[", RowBox[{"S", "[", RowBox[{"3", ",", "2", ",", "1", ",", "n"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.949485098212232*^9, 3.949485110224143*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"5c2c201f-3679-4950-8827-1ead90a4c342"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "4"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "3"], "=", "1"}], SubscriptBox["\[Omicron]", "4"]], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], SubscriptBox["\[Omicron]", "3"]], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], SubsuperscriptBox["\[Omicron]", "3", "2"]]}], SubsuperscriptBox["\[Omicron]", "4", "3"]]}]], "Output", CellChangeTimes->{3.949641109117264*^9}, CellLabel->"Out[10]=",ExpressionUUID->"ac47ca83-d2bd-49c0-8228-55b02f4c087c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SExpansion", "[", RowBox[{ RowBox[{"S", "[", RowBox[{"3", ",", "2", ",", "1", ",", "n"}], "]"}], ",", "n", ",", "6"}], "]"}]], "Input", CellChangeTimes->{{3.9494850398117*^9, 3.949485052421691*^9}, { 3.9494851290386677`*^9, 3.949485129913015*^9}}, CellLabel->"In[11]:=",ExpressionUUID->"e37c426f-a94a-4f48-8198-9a842ee9fd29"], Cell[BoxData[ RowBox[{ FractionBox["17", RowBox[{"48", " ", SuperscriptBox["n", "6"]}]], "+", FractionBox["107", RowBox[{"1800", " ", SuperscriptBox["n", "5"]}]], "-", FractionBox["15", RowBox[{"32", " ", SuperscriptBox["n", "4"]}]], "+", FractionBox["4", RowBox[{"9", " ", SuperscriptBox["n", "3"]}]], "-", FractionBox[ RowBox[{"143", " ", SuperscriptBox["z2", "3"]}], "210"], "+", RowBox[{ RowBox[{"(", RowBox[{ FractionBox["1", RowBox[{"6", " ", SuperscriptBox["n", "6"]}]], "-", FractionBox["1", RowBox[{"2", " ", SuperscriptBox["n", "4"]}]], "+", FractionBox["1", SuperscriptBox["n", "3"]], "-", FractionBox["1", SuperscriptBox["n", "2"]]}], ")"}], " ", "z3"}], "+", RowBox[{"3", " ", SuperscriptBox["z3", "2"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox["7", RowBox[{"24", " ", SuperscriptBox["n", "6"]}]]}], "+", FractionBox["37", RowBox[{"60", " ", SuperscriptBox["n", "5"]}]], "-", FractionBox["5", RowBox[{"8", " ", SuperscriptBox["n", "4"]}]], "+", FractionBox["1", RowBox[{"3", " ", SuperscriptBox["n", "3"]}]]}], ")"}], " ", RowBox[{"LG", "[", "n", "]"}]}]}]], "Output", CellChangeTimes->{3.9496411094651012`*^9}, CellLabel->"Out[11]=",ExpressionUUID->"80556b9e-b7a0-484c-bdd0-0d36f9934374"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The double sum", "Subsection",ExpressionUUID->"75828a7d-5583-41ac-8a6b-88b7aeea3b74"], Cell[BoxData[ RowBox[{ RowBox[{"doubleSum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "a"], RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]], "+", FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]], "-", FractionBox[ RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"k", "+", "n"}]}], "]"}], RowBox[{"k", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}]}]]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{ 3.5265792442966394`*^9, {3.949319405851594*^9, 3.9493194068370533`*^9}, { 3.9493195033876677`*^9, 3.949319507495893*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"67b8a565-abde-4512-b681-30018ea1dd36"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rec", "=", RowBox[{ RowBox[{"GenerateRecurrence", "[", RowBox[{"doubleSum", ",", "n", ",", RowBox[{"SimplifyByExt", "->", "None"}]}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.5265792538361855`*^9, 3.5265792646308026`*^9}, 3.9493195085529537`*^9}, CellLabel->"In[13]:=",ExpressionUUID->"2b6aa7cf-db86-40c1-a159-f75298f0ce5f"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"n", " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", "n"}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}]}], "\[Equal]", RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "a"}], ")"}], " ", RowBox[{"(", RowBox[{"a", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "a"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"a", "+", "n"}]}], "]"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"], " ", RowBox[{"(", RowBox[{"1", "+", "a", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "a", "+", "n"}], ")"}]}], ")"}]}], ")"}]}]}]], "Output", CellChangeTimes->{3.949641111482437*^9}, CellLabel->"Out[13]=",ExpressionUUID->"439f749c-1f89-44e5-b247-20c28b3b8957"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"rec", "=", RowBox[{"LimitRec", "[", RowBox[{"rec", ",", RowBox[{"SUM", "[", "n", "]"}], ",", RowBox[{"{", "n", "}"}], ",", "a"}], "]"}]}]], "Input", CellChangeTimes->{{3.560493744246934*^9, 3.5604937453919992`*^9}, { 3.949319515797727*^9, 3.949319519781988*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"37427790-8ecf-47d0-96f0-49b8e144a929"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"n", " ", RowBox[{"SUM", "[", "n", "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}]}], "\[Equal]", RowBox[{"-", FractionBox[ RowBox[{"1", "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]]}]}]], "Output", CellChangeTimes->{3.949641111802196*^9}, CellLabel->"Out[14]=",ExpressionUUID->"e3376e21-0c22-436e-a6f4-fe395704f35a"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recSol", "=", RowBox[{"SolveRecurrence", "[", RowBox[{"rec", ",", RowBox[{"SUM", "[", "n", "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.949319524900738*^9, 3.9493195254470177`*^9}}, CellLabel->"In[15]:=",ExpressionUUID->"68cd39e3-2a5d-4760-b9f4-6b12d57ae7d4"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", FractionBox["1", RowBox[{"n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]]}]}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{ FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], "2"], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.9496411120969133`*^9}, CellLabel->"Out[15]=",ExpressionUUID->"f0ee9114-c891-4edd-a554-60030241dc51"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindLinearCombination", "[", RowBox[{"recSol", ",", RowBox[{"{", RowBox[{"1", ",", RowBox[{"{", RowBox[{"1", "/", "2"}], "}"}]}], "}"}], ",", "n", ",", "1"}], "]"}]], "Input", CellChangeTimes->{{3.949319529152038*^9, 3.9493195296298018`*^9}}, CellLabel->"In[16]:=",ExpressionUUID->"025d336f-d5ea-4719-b236-fde3f43cea1d"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], "2"], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]]}]], "Output", CellChangeTimes->{3.949641112124337*^9}, CellLabel->"Out[16]=",ExpressionUUID->"073e7f4c-bf87-45cb-8e8b-c7fa597633e2"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Solving of large recurrences", "Section", CellChangeTimes->{{3.695359139537485*^9, 3.695359158805407*^9}, { 3.878618044580777*^9, 3.878618045338767*^9}},ExpressionUUID->"e9d8a45e-77a3-4dbd-8ffe-\ ffe386a4f390"], Cell[BoxData[ RowBox[{ RowBox[{"rec", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"2419791630618001724349719825822515200", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"46893058834334803550048105380734566400", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"442193057465681392730605627834886258688", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"2705163824287918413734939189834632593408", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"12077169282974589920658629516473115295744", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"41966837290479779427524408810546219827200", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"118219101947163710354472195018903784452096", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"277636548018756319924282582437965801250816", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"554818694316590989606680673486591181169664", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"958215852325427069871263145576988323096576", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"1447807538463763344480353057138198170764544", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"1932634574589957327485740054131528807341312", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"2297546075527773923214701010279556037859200", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"2448775269439910075996659926897517474168064", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"2353082458409861559967790811417795090328800", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"2048299114135230176735374555025680366259616", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"1621741955236083712069918941593026341937328", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"1171973077100571998520049114011408940864324", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"775366789376058136594755591722578221832754", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"470845228207878464545078102494550968145626", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"263029261282480708488833600406560031424266", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"135432625185170494152646639679878430576366", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"64380525008058346133125819015855307039890", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"28294941869896718238659823138529447643409", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"11510639826509868241332665511273019883245", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"4338601127644953578265748144467101651566", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"1516347765344935407227461978628652931140", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"491707203871221783491014552667264654348", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"147998315309037544218073839785461595524", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"41357982388489608344142528638546419812", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"10731279186864834895938990612954501362", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"2585221832059644562908138564950975983", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"578081901021769208062617955849607255", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"119933836608195636314895593780004412", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"23072349655406047775407594541682536", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"4112383801613288221238341717431870", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"678441475758950513418671792499790", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"103470631338396383215642368754950", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"14567035475277568652763736374038", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"1889820701704575278082350304095", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"225464981620744825055854151899", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"24677977218000736361042595318", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"2471158501904322420447817484", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"225650752266000765364410056", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"18718039227261648047794832", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"1404175763444631451277224", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"94757256722843508771238", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"5715830429727778230433", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"305843846362020680945", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"14381720253879875868", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"587432077055797654", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"20535798664117844", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"602646805448372", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"14460796607552", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"273265680480", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"3838554144", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"36289696", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"187328", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"256", " ", SuperscriptBox["n", "61"]}]}], ")"}]}], " ", RowBox[{"f", "[", "n", "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"2184637011646089260832263446242263040", "+", RowBox[{"60213944484149858917493971997278863360", " ", "n"}], "+", RowBox[{"765667472926191423796966419188114522112", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"6105365080510451396982635521026428239872", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"34708860405036002296270093362827805720576", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"151012015780667833836461483157418876403712", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"525927125380365588824032094032506978336768", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"1512196534433079466364883142044247431979008", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"3671470718711877962933422750534015631462400", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"7656732454200652158809870797561463793930240", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"13900418857159591484188788012697295475566592", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"22205000857672892951155779495290643697085440", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"31486034126294839958942742570718414414535680", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"39919692863067272123801901519465590396631552", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"45531432069748835165613722710992028682930304", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"46961436588813572025777511789038901263777888", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"43994959907297109868687362235336122063739280", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"37579715685313851159102021797411799861122656", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"29364773248686683915239812679461616136356200", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"21050691034785332915492476057834318260647992", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"13878863958177278906363320901764654180674068", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"8433920328583831248209974031887597144856166", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"4732703540813596136660683547266937537829654", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"2456402281573059079053298938442251846697230", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"1180884360800419306100550729721645208294214", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"526441908393454455371612300421050477868182", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"217853112783218022232146626128806211983463", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"83754020847842024722911201861265143892515", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"29933835570003312829692764177995129463678", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"9950649553145001634744062228609687611636", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"3077685237402338181588673390623771087768", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"885864683981666005570651087823908853468", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"237303300491689480050502026965768922040", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"59155090124648479353044037293149067126", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"13719117466936551766792191652359249405", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"2958917780333050813266891451512650209", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"593157044864289496291653275756884512", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"110438535500745724875815791275373824", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"19080579219381871689558731310265194", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"3055650762530616827066946087618154", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"452987258893586741202300418715142", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"62067608935535379147520211092898", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"7846086266834729051216715205681", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"913130773095847812486509019205", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"97597613016813524555747415182", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"9552877857235409775526229148", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"853450995298842653501332876", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"69325405419294490888090944", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"5096893257290771080869644", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"337358454813758704464466", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"19974662258188401941987", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"1049859496149284305279", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"48525482667944885632", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"1949521858753528714", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"67075940936545308", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"1938510016023020", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"45836200086880", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"854037611552", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"11836382688", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"110495776", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"564032", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"768", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "22022314605592690531751005935108096000"}], "-", RowBox[{"289135277001687902353264231969613414400", " ", "n"}], "-", RowBox[{"1488037313005848582765208942457703628800", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"1818726834376404926186570730184369766400", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"23382204546977767384985015673369488916480", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"188082685608246892321802873635659960877056", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"825801477256291595460486042061384946712576", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"2628541026330989080112440997321895434330112", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"6629655427580241497225184251016863653908480", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"13848216502549635881680294337223876185911296", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"24611563389512504073584247446699535922575360", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"37909511298937488708854223877932503102176256", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"51299918554927297426860529427157650519514112", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"61629468272462855574906800764536339189128192", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"66280124475672744973400889288785644773739136", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"64247409978580992116427545385119399679833312", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"56449472002389683597707337063406362546728272", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"45171322891361752871580719683909268264681440", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"33053824818470328765890536318636510140913496", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"22194286968981437345392564520988385907626416", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"13715606341403352934357761159783608231286212", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"7820919263011960290532633923574013757937322", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"4124101824382320133598766513489500768514642", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"2014900658274015221853754922074596765809754", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"913552367745160482379609090421491101869914", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"384912898173545720900056698586842503122918", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"150881012199256958147419290021749103219985", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"55074624810472925390185481773716375577417", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"18733933409151309041626750964983474454358", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"5941570334258463328802562879066834157404", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"1757621107214989234449316717174814410560", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"485046390298804579275932040902973332692", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"124877689424959601610199276249591868356", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"29989412247749884342281895499180335754", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"6715821939958386382569103248806082447", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"1401758065674213210274260438674133863", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"272527587406079568538323090035115684", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"49312358430897053286589310725757704", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"8296078956189808509997550437883390", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"1296103778071525102182189470635774", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"187776581998229163368113485479490", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"25186081309681650194834084101162", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"3121520966967059012579790235663", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"356696551136881314117558024007", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"37484536898709537869638087550", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"3612012600567270037248636788", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"318063209276420217186951276", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"25493505002286210193173232", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"1851374597173806436884944", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"121157534952438954351366", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"7099015219827843837177", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"369549841897016581817", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"16930667101318371492", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"674702033398352670", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"23042430467695396", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"661436747087492", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"15543736473056", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"288011538912", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"3971907936", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"36920096", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"187840", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"256", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"2", "+", "n"}], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"27276878942915969107316087331005399040", "+", RowBox[{"705128506368571028133010447867324661760", " ", "n"}], "+", RowBox[{"8795984422030468128177510606635859443712", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"69923155581931005683616191106190820769792", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"397777067188645895439894469915116729532416", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"1729204532204996997152583407397237453225984", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"5995525164632894191922695828861346535505920", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"17088735819395764913747809298043883399888896", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"40950242468668902568149188809126378671144960", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"83944922396996847271825479812337521047842816", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"149241076173129905329165504863175575091332096", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"232683501679364482345805030869544077599268352", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"321069920926096778719573512107074163678806528", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"395099424821926843732271053703633605285841664", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"436399473903028869076842967038797890370004032", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"435026967213159405048515684405715110378338720", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"393230111147319510033909063747616966700500160", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"323623592515063057301937486575186179314548656", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"243346679569443698265032683268250796082355872", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"167699480512372076874814970574805341647426008", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"106198031145430172090232700406650650176362252", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"61942356363090378267431321236277379702842498", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"33344350577247386451113806267465360943413242", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"16595113813714428964861691068549243862900514", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"7647457411737976747992069047332333397916774", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"3267331144899381542768858934068187529943662", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"1295632087778176811117167430324892967262289", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"477281953036655577658172066157254622874249", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"163451731801877305191319996120051969522930", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"52067844190687573163358660941762172198124", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"15434420976233983819168350126595595202140", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"4258528038477820285037384293734708124148", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"1093751039753805052074448772782500867240", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"261482348137970277384591678834281844450", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"58175398225026023823129376451604957051", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"12040624671354180854266520205844399991", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"2317073223658459084340945199597363112", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"414290751988803620905542484937609472", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"68763751517799771571460168482707342", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"10583544268756001126111015357786438", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"1508533267822499512906498412337430", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"198821183372782469752192989049026", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"24186388725302176057341026698695", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"2709982082198798782794681162855", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"278988479854541096081379081954", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"26314522078796930198189306612", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"2266504237219506659730716168", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"177578797780695304231916832", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"12598833333382387109147196", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"805098410198134438811902", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"46044417764504993310349", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"2338717637745860215073", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"104514427357536583000", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"4061699225871964542", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"135250177600636980", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"3784921061092388", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"86707444117568", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"1566227411552", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"21060262112", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"190971744", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"949440", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"1280", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"3", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "167707728739563162662793640641822720000"}], "-", RowBox[{"1884621099579099264283288794700972032000", " ", "n"}], "-", RowBox[{"5041377848806910695366548334073767526400", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"46775477320896085249578292645052798730240", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"556885835525484915713810095094705435443200", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"3159242452468551228562153543444819393904640", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"12415541071618540660307393273559673826377728", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"37637925296940019781324185478731922295963648", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"92737714460401157604501008974332095412830208", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"191667000496318490779014701563244967355920384", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"339452263097369421820859975846282755107529728", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"523248300281830339242284804346885285914676736", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"710386898168009760958599932526536214775457280", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"857462533335657188694551242676693238363818752", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"927185892315490057264223661958052089087716416", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"903794264491665943887994150618261070781888544", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"798358468088992068216784561678832305544914688", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"641910993104210400936977369822714460830564208", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"471562530047346319754202040007262652757218624", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"317538797128843618565893715748339861357902640", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"196542800425272936030894969376573161178781716", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"112089205036988299699706179845387148374717002", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"59022381396427738240092380898572083968461562", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"28746865359106416256416247321136578512534282", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"12970254953319120480983509703101810054182782", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"5428183614772190906005960866663410280479562", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"2109510758277426773606416875119872314811877", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"761938745869645201813190530563834432295465", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"255965695136511737849398726940189624590934", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"80021356179793507587079440531706016662668", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"23289571211370481847378953662633197929068", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"6311748066033135567176717489354950656844", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"1592960015056501397895148828890424971668", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"374365028306562451548967569889349299994", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"81907187008712905581886627733322657947", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"16676994045570734948021794434982584627", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"3158233696398401126430392498360079036", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"555889720171821324468177471226088216", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"90857083052539834834050188828261198", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"13774542091612157361488698113634638", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"1934519485240215372599275594201750", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"251288787124077923597876350839934", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"30136222767569363668692805964139", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"3329676213071448715046083817455", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"338100507320636683862629994494", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"31461631160467405736547997940", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"2674035177086832347621329096", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"206786377638960225950374136", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"14483513059358524937574360", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"913893080887697869410574", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"51619555838689353242549", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"2589956035399262629893", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"114354608201432199836", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"4391706562159684062", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"144543224194970180", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"3998894202305188", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"90585200571648", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"1618378146784", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"21530125088", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"193247264", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"952000", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"1280", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"4", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"146973883013564258890458767567093760000", "+", RowBox[{"2229259984516059314164243577955680256000", " ", "n"}], "+", RowBox[{"16595245522055651232438964274595574579200", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"82110213857892331021373929579308684410880", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"310740084394832008017420625315940985077760", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"983450262875422499074965195835230498914304", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"2752795738160705474897634293607251897352192", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"6979426901876915027726954745414540323635200", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"16022042987367137971152393882056321496649728", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"32976239563629576537093753171267079363952640", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"60348223112048685326140378332733677958551552", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"97870234277454774181867539360398981127882752", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"140769737681664961820732259956248222002156544", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"180159949407962743950515756408355287251632128", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"206063708436712977188583276398745589001139968", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"211629972252724984409247015325612755050864384", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"196052177074481560658330639762342332344444048", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"164525447978078562567838918443591164707502416", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"125556179875444843058702170521611753698519256", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"87435155865587777892765113376420118121960328", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"55732185112640160274945213210191691809530492", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"32603848226154254694224926180497667742692898", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"17546915408521547104784732751147391512437658", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"8705564957790628615250340784589164051237410", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"3988719282955627082164451161346964412614922", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"1690355246982168411514553093925250510197586", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"663435741557456861450332956793381788403741", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"241419754068235849516846929157324919604649", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"81525053083740624437032057330999824995610", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"25566108133626930799108150870844970750372", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"7449510928708389104237268058331083573272", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"2017628218981886286590169884480431538092", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"508034801230647487942634941743803660008", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"118932982740371640583748836229570720114", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"25882966091402668443156859423413630159", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"5234831252327150258495816833335987083", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"983485163689520917457934207647204592", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"171526535385991276074256199958715648", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"27748080934509521703257731932900950", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"4159340594584083287370449291142406", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"576981874549508051423551767867114", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"73960066614630153761116279570486", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"8745090271861061856731036132491", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"951844347105863253364908309823", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"95138142685999346905079417738", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"8707815519389802473929517372", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"727452758910755330250630756", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"55255418993869721987243656", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"3798932121017318727211124", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"235152612408584314355478", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"13022051437733827813521", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"640211109261103389509", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"27683156067422637168", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"1040643976182868542", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"33508905375528340", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"906557733231620", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"20073465051104", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"350427896160", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"4554253344", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"39936608", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"192448", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"256", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"5", "+", "n"}], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "229133653589293103588089989773131776000"}], "-", RowBox[{"2707218686086927844219339121202456166400", " ", "n"}], "-", RowBox[{"8435158839130133492119907321120544522240", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"57312106282972020092113166164705144995840", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"761311556185420846905694442317899022860288", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"4470070472792421402760998791768772851073024", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"17933275241733080753971051500884064151535616", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"55163890293619386176182666327824835758047232", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"137396641381867437272033940971117581338464256", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"286236922551698791964818718459230151232184320", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"509799264887681665018259066709529133495617536", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"788651555674921555121999082128867307071072256", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"1072588143491400565323822921025679581406329856", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"1294744188580567907082105180269849560142171648", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"1397941057071523729674966499885688914786767104", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"1358660215905239775278697600361668905916384384", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"1194983627524612608767092393372811905774530512", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"955428594075553278209673752244502528652871632", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"697089659895187705783120568296945401635556904", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"465656730189159752034253308339764890945988496", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"285604431687495229880881251511616073109650220", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"161232532865686268214919319101899922033636862", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"83955845991827386122821327404717592026870110", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"40397529485287635174511174996855520331201558", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"17990657209554183341396289732046387408087174", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"7425224905641533823522969250408974792884434", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"2843367741942140922093698660861113923900963", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"1011168561548543796801655543741138811011507", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"334201980134186841985596581694146281914418", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"102717210317860109532509242812022564194860", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"29370354351546973498662057262843294604096", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"7814863828387728382126093354840716158868", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"1935205500844458362351907362574701252076", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"445970814156252687541717555256393051710", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"95625156036928688663360232186306822077", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"19070756523821114762082917501786328629", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"3535622715616424883553216938987081532", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"608924288966351008440912039102146104", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"97336892660236397648749897843495650", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"14425803764696881833933919196754354", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"1979652359568410734625780917990814", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"251164938768336941770845524474110", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"29408294486395333803864974921629", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"3171106066171398274360702357901", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"314139274560185545478142702954", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"28508357961055344546198016916", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"2362252408109032902335223380", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"178036706377314851710827640", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"12149450550976441759660448", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"746700160166119020800866", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"41068868731260336676891", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"2005971167965587366107", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"86201183906778839132", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"3221230995223510058", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"103140014798676940", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"2775465779235692", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"61145713649888", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"1062388816288", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"13746710944", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"120074336", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"576832", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"768", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"6", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "205329880537617010501022298748747776000"}], "-", RowBox[{"2696078286141154627238041660556732006400", " ", "n"}], "-", RowBox[{"13446746876280445907576988016652310282240", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"9572101437141628816667721484850492866560", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"280182940654525918506623162238291749634048", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"2094095152538489412154535455992036382212096", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"9063043112286942735738908649797197564280832", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"28763695575379100484547551886516377655984128", " ", SuperscriptBox["n", "7"]}], "+", RowBox[{"72587277948926887578249012833148937342402560", " ", SuperscriptBox["n", "8"]}], "+", RowBox[{"151848308039043719353760639271736941736587264", " ", SuperscriptBox["n", "9"]}], "+", RowBox[{"270228635931765132109743959410128481769127936", " ", SuperscriptBox["n", "10"]}], "+", RowBox[{"416475851930582110790956894347255568214041600", " ", SuperscriptBox["n", "11"]}], "+", RowBox[{"563273528848780505801853290651985532735597568", " ", SuperscriptBox["n", "12"]}], "+", RowBox[{"675378713606734164964894487190643586775823616", " ", SuperscriptBox["n", "13"]}], "+", RowBox[{"723772257202863431434275435199614271966064896", " ", SuperscriptBox["n", "14"]}], "+", RowBox[{"697846465839303446043836853429870244915884800", " ", SuperscriptBox["n", "15"]}], "+", RowBox[{"608706380872990973730902890434221992542828992", " ", SuperscriptBox["n", "16"]}], "+", RowBox[{"482560435172785135279241439251298345923241248", " ", SuperscriptBox["n", "17"]}], "+", RowBox[{"349054980225759116262635139230522365632991808", " ", SuperscriptBox["n", "18"]}], "+", RowBox[{"231147595596532620592379457194557347102467288", " ", SuperscriptBox["n", "19"]}], "+", RowBox[{"140536192755338951691788258703532233289176732", " ", SuperscriptBox["n", "20"]}], "+", RowBox[{"78644201689438279267336949795701480453983370", " ", SuperscriptBox["n", "21"]}], "+", RowBox[{"40593143275902089690541181091180591688214746", " ", SuperscriptBox["n", "22"]}], "+", RowBox[{"19361740620088456118995567543717926580867170", " ", SuperscriptBox["n", "23"]}], "+", RowBox[{"8547241017147324196672102915309670663358310", " ", SuperscriptBox["n", "24"]}], "+", RowBox[{"3496868703574705660432884441070148864445574", " ", SuperscriptBox["n", "25"]}], "+", RowBox[{"1327376171171369752308048746304225126827565", " ", SuperscriptBox["n", "26"]}], "+", RowBox[{"467923258436459092038397095600958333553789", " ", SuperscriptBox["n", "27"]}], "+", RowBox[{"153301662682604734680685528474464248493050", " ", SuperscriptBox["n", "28"]}], "+", RowBox[{"46704858741585641410966274980319548456852", " ", SuperscriptBox["n", "29"]}], "+", RowBox[{"13237336261385959202565843547692916281036", " ", SuperscriptBox["n", "30"]}], "+", RowBox[{"3491195603476702931319945067604568162828", " ", SuperscriptBox["n", "31"]}], "+", RowBox[{"856894095632874258492412142645342853768", " ", SuperscriptBox["n", "32"]}], "+", RowBox[{"195720752557255112678769676790740646042", " ", SuperscriptBox["n", "33"]}], "+", RowBox[{"41592327466920617640131171642573300047", " ", SuperscriptBox["n", "34"]}], "+", RowBox[{"8220478083640342014901813382323247819", " ", SuperscriptBox["n", "35"]}], "+", RowBox[{"1510281458336947632567400840635396600", " ", SuperscriptBox["n", "36"]}], "+", RowBox[{"257744957982899404990201966847018128", " ", SuperscriptBox["n", "37"]}], "+", RowBox[{"40823363729408037724579306410090910", " ", SuperscriptBox["n", "38"]}], "+", RowBox[{"5994359838556501099719702325281158", " ", SuperscriptBox["n", "39"]}], "+", RowBox[{"814944222044225628890907403778678", " ", SuperscriptBox["n", "40"]}], "+", RowBox[{"102422776251835173864123492495018", " ", SuperscriptBox["n", "41"]}], "+", RowBox[{"11878625574103989575227584857771", " ", SuperscriptBox["n", "42"]}], "+", RowBox[{"1268598347854663415431294550179", " ", SuperscriptBox["n", "43"]}], "+", RowBox[{"124454005507987360466745640362", " ", SuperscriptBox["n", "44"]}], "+", RowBox[{"11183726605231249216080033340", " ", SuperscriptBox["n", "45"]}], "+", RowBox[{"917531675574530213244008152", " ", SuperscriptBox["n", "46"]}], "+", RowBox[{"68459846335433575358003800", " ", SuperscriptBox["n", "47"]}], "+", RowBox[{"4624495807310652011314684", " ", SuperscriptBox["n", "48"]}], "+", RowBox[{"281309213929495034504854", " ", SuperscriptBox["n", "49"]}], "+", RowBox[{"15311862610709369056793", " ", SuperscriptBox["n", "50"]}], "+", RowBox[{"740055853160834519261", " ", SuperscriptBox["n", "51"]}], "+", RowBox[{"31464751682713946856", " ", SuperscriptBox["n", "52"]}], "+", RowBox[{"1163191934453543606", " ", SuperscriptBox["n", "53"]}], "+", RowBox[{"36840351024083332", " ", SuperscriptBox["n", "54"]}], "+", RowBox[{"980508441304116", " ", SuperscriptBox["n", "55"]}], "+", RowBox[{"21362825138432", " ", SuperscriptBox["n", "56"]}], "+", RowBox[{"367049792736", " ", SuperscriptBox["n", "57"]}], "+", RowBox[{"4696674016", " ", SuperscriptBox["n", "58"]}], "+", RowBox[{"40576224", " ", SuperscriptBox["n", "59"]}], "+", RowBox[{"192960", " ", SuperscriptBox["n", "60"]}], "+", RowBox[{"256", " ", SuperscriptBox["n", "61"]}]}], ")"}], " ", RowBox[{"f", "[", RowBox[{"7", "+", "n"}], "]"}]}]}], "\[Equal]", "0"}]}], ";"}]], "Input", CellLabel->"In[17]:=",ExpressionUUID->"36d1a0d5-e27b-46eb-bdc6-e5c576af20bb"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recSol", "=", RowBox[{"SolveRecurrence", "[", RowBox[{"rec", ",", RowBox[{"f", "[", "n", "]"}], ",", RowBox[{"IndefiniteSummation", "->", "False"}], ",", RowBox[{"SigmaPrint", "\[Rule]", "True"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.550748961274764*^9, 3.5507489660980406`*^9}, { 3.5507490483227434`*^9, 3.5507490515729294`*^9}, {3.5720029116034946`*^9, 3.572002913005575*^9}, {3.6780889735800333`*^9, 3.678088978870099*^9}, { 3.695385457301876*^9, 3.695385459179861*^9}, {3.878474302212434*^9, 3.878474307767787*^9}}, CellLabel->"In[18]:=",ExpressionUUID->"5adadb75-9c29-49e4-8c9b-a5f2e82dcda2"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "7"}], SequenceForm["Solve recurrence of order: ", 7], Editable->False]], "Print", CellChangeTimes->{3.949641112269651*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"d8854948-3c7d-4c05-978b-f5ba8aab79f6"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "6"}], SequenceForm["Solve recurrence of order: ", 6], Editable->False]], "Print", CellChangeTimes->{3.949641113313389*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"462e58a7-ce2d-4f14-8bc2-689ed6d05e7e"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "5"}], SequenceForm["Solve recurrence of order: ", 5], Editable->False]], "Print", CellChangeTimes->{3.949641114597748*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"a1b49246-6849-4ba7-a424-b0585fe8c997"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "4"}], SequenceForm["Solve recurrence of order: ", 4], Editable->False]], "Print", CellChangeTimes->{3.949641114689477*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"95f456c4-90a9-4f62-a51d-f69a1e09ad94"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "3"}], SequenceForm["Solve recurrence of order: ", 3], Editable->False]], "Print", CellChangeTimes->{3.949641114809656*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"5db666da-b813-493f-a1f3-6ba23a565104"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "2"}], SequenceForm["Solve recurrence of order: ", 2], Editable->False]], "Print", CellChangeTimes->{3.949641116230908*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"efbebf76-651f-41b1-bb1f-4fa131bc8d37"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "1"}], SequenceForm["Solve recurrence of order: ", 1], Editable->False]], "Print", CellChangeTimes->{3.9496411200862503`*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"e088f4c7-0697-4f27-9319-1242fbfdd5cb"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "0"}], SequenceForm["Solve recurrence of order: ", 0], Editable->False]], "Print", CellChangeTimes->{3.949641121948248*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"25785fe4-4c61-485a-885e-fb4ee2ddef5b"], Cell[BoxData["\<\"Simplification of solutions.\"\>"], "Print", CellChangeTimes->{3.949641121968378*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"514792c9-efdb-48b1-94dd-04332a09fa75"], Cell[BoxData["\<\"Computation of lower bounds.\"\>"], "Print", CellChangeTimes->{3.949641122347658*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"bf00d469-4116-4d96-b2b6-f8cac17fcd36"], Cell[BoxData["\<\"Interpretation of solutions as sums and products.\"\>"], \ "Print", CellChangeTimes->{3.949641122642805*^9}, CellLabel-> "During evaluation of \ In[18]:=",ExpressionUUID->"c2b5f774-c508-436f-9987-90b20453ca39"] }, Open ]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "4"}], "n"], FractionBox[ RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "1"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "+", SubsuperscriptBox["\[Iota]", "1", "4"]}], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "4"}], "n"], FractionBox[ RowBox[{ RowBox[{"-", "68"}], "-", RowBox[{"23", " ", SubscriptBox["\[Iota]", "1"]}], "+", RowBox[{"37", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "-", RowBox[{"66", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "-", RowBox[{"13", " ", SubsuperscriptBox["\[Iota]", "1", "4"]}], "+", SubsuperscriptBox["\[Iota]", "1", "5"]}], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}]]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "4"}], "n"], FractionBox[ RowBox[{"424", "+", RowBox[{"146", " ", SubscriptBox["\[Iota]", "1"]}], "-", RowBox[{"253", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "+", RowBox[{"396", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "1", "4"]}], "-", RowBox[{"14", " ", SubsuperscriptBox["\[Iota]", "1", "5"]}], "+", SubsuperscriptBox["\[Iota]", "1", "6"]}], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "12"}], "n"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "1"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "+", SubsuperscriptBox["\[Iota]", "1", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "10"}], SubscriptBox["\[Iota]", "1"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "2"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "2", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "2", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "2", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "2", "7"]}], "+", SubsuperscriptBox["\[Iota]", "2", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "3"], "=", "8"}], SubscriptBox["\[Iota]", "2"]], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "3"]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "+", SubsuperscriptBox["\[Iota]", "3", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "3", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "9"]}], "+", SubsuperscriptBox["\[Iota]", "3", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "4"], "=", "6"}], SubscriptBox["\[Iota]", "3"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", SubsuperscriptBox["\[Iota]", "4", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1036800"}], "+", RowBox[{"50108544", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"378482688", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"1203879360", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1866851440", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"901685472", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "+", RowBox[{"1958467456", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "-", RowBox[{"4787603188", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "+", RowBox[{"5528964798", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"4152500203", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", RowBox[{"2141149820", " ", SubsuperscriptBox["\[Iota]", "4", "10"]}], "-", RowBox[{"715311261", " ", SubsuperscriptBox["\[Iota]", "4", "11"]}], "+", RowBox[{"98930622", " ", SubsuperscriptBox["\[Iota]", "4", "12"]}], "+", RowBox[{"42122506", " ", SubsuperscriptBox["\[Iota]", "4", "13"]}], "-", RowBox[{"32513904", " ", SubsuperscriptBox["\[Iota]", "4", "14"]}], "+", RowBox[{"11176602", " ", SubsuperscriptBox["\[Iota]", "4", "15"]}], "-", RowBox[{"2396078", " ", SubsuperscriptBox["\[Iota]", "4", "16"]}], "+", RowBox[{"327953", " ", SubsuperscriptBox["\[Iota]", "4", "17"]}], "-", RowBox[{"26348", " ", SubsuperscriptBox["\[Iota]", "4", "18"]}], "+", RowBox[{"935", " ", SubsuperscriptBox["\[Iota]", "4", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "20"]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SubsuperscriptBox["\[Iota]", "4", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "7200"}], "+", RowBox[{"53760", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"130880", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"160564", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"115948", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"52522", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"15019", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"2548", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"202", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "4"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}], ")"}]}]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "12"}], "n"], RowBox[{ FractionBox["1", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}]], RowBox[{ RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "1"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "+", SubsuperscriptBox["\[Iota]", "1", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "10"}], SubscriptBox["\[Iota]", "1"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "2"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "2", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "2", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "2", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "2", "7"]}], "+", SubsuperscriptBox["\[Iota]", "2", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "3"], "=", "8"}], SubscriptBox["\[Iota]", "2"]], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "3"]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "+", SubsuperscriptBox["\[Iota]", "3", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "3", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "9"]}], "+", SubsuperscriptBox["\[Iota]", "3", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "4"], "=", "6"}], SubscriptBox["\[Iota]", "3"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", SubsuperscriptBox["\[Iota]", "4", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1036800"}], "+", RowBox[{"50108544", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"378482688", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"1203879360", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1866851440", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"901685472", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "+", RowBox[{"1958467456", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "-", RowBox[{"4787603188", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "+", RowBox[{"5528964798", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"4152500203", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", RowBox[{"2141149820", " ", SubsuperscriptBox["\[Iota]", "4", "10"]}], "-", RowBox[{"715311261", " ", SubsuperscriptBox["\[Iota]", "4", "11"]}], "+", RowBox[{"98930622", " ", SubsuperscriptBox["\[Iota]", "4", "12"]}], "+", RowBox[{"42122506", " ", SubsuperscriptBox["\[Iota]", "4", "13"]}], "-", RowBox[{"32513904", " ", SubsuperscriptBox["\[Iota]", "4", "14"]}], "+", RowBox[{"11176602", " ", SubsuperscriptBox["\[Iota]", "4", "15"]}], "-", RowBox[{"2396078", " ", SubsuperscriptBox["\[Iota]", "4", "16"]}], "+", RowBox[{"327953", " ", SubsuperscriptBox["\[Iota]", "4", "17"]}], "-", RowBox[{"26348", " ", SubsuperscriptBox["\[Iota]", "4", "18"]}], "+", RowBox[{"935", " ", SubsuperscriptBox["\[Iota]", "4", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "20"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "5"], "=", "3"}], SubscriptBox["\[Iota]", "4"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "7200"}], "+", RowBox[{"53760", " ", SubscriptBox["\[Iota]", "5"]}], "-", RowBox[{"130880", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "+", RowBox[{"160564", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "-", RowBox[{"115948", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "+", RowBox[{"52522", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "-", RowBox[{"15019", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "+", RowBox[{"2548", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "-", RowBox[{"202", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "+", SubsuperscriptBox["\[Iota]", "5", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "268061736960"}], "+", RowBox[{"2824243780608", " ", SubscriptBox["\[Iota]", "5"]}], "+", RowBox[{"4318123156992", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "-", RowBox[{"201123626821632", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "+", RowBox[{"1391666715407232", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "-", RowBox[{"5277178048925952", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"12962808461111200", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"21305292915332800", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"21166111959705696", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"2890677091472172", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "-", RowBox[{"33448944651882246", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "+", RowBox[{"73993322390651126", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "-", RowBox[{"99697525001566438", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"100147841927725059", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"79546250838626709", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"50955754267106775", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"26260396787425091", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"10563977846009515", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"2989102604911489", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"315718564141377", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"242070039538587", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}], "-", RowBox[{"192965061962541", " ", SubsuperscriptBox["\[Iota]", "5", "21"]}], "+", RowBox[{"85146456468263", " ", SubsuperscriptBox["\[Iota]", "5", "22"]}], "-", RowBox[{"27671025884951", " ", SubsuperscriptBox["\[Iota]", "5", "23"]}], "+", RowBox[{"7076120002347", " ", SubsuperscriptBox["\[Iota]", "5", "24"]}], "-", RowBox[{"1452474125119", " ", SubsuperscriptBox["\[Iota]", "5", "25"]}], "+", RowBox[{"239656119375", " ", SubsuperscriptBox["\[Iota]", "5", "26"]}], "-", RowBox[{"31463978263", " ", SubsuperscriptBox["\[Iota]", "5", "27"]}], "+", RowBox[{"3217398027", " ", SubsuperscriptBox["\[Iota]", "5", "28"]}], "-", RowBox[{"247073926", " ", SubsuperscriptBox["\[Iota]", "5", "29"]}], "+", RowBox[{"13382758", " ", SubsuperscriptBox["\[Iota]", "5", "30"]}], "-", RowBox[{"452800", " ", SubsuperscriptBox["\[Iota]", "5", "31"]}], "+", RowBox[{"6936", " ", SubsuperscriptBox["\[Iota]", "5", "32"]}], "+", RowBox[{"16", " ", SubsuperscriptBox["\[Iota]", "5", "33"]}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "5"], "."]]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"14892318720", "-", RowBox[{"139725950976", " ", SubscriptBox["\[Iota]", "5"]}], "+", RowBox[{"544478712960", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "-", RowBox[{"1213653772800", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "+", RowBox[{"1752329541504", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "-", RowBox[{"1727022580784", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"1160998559536", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"479387084584", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"42444457372", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "+", RowBox[{"101229213030", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "-", RowBox[{"88422557953", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "+", RowBox[{"43907846563", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "-", RowBox[{"15419001366", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"4044155236", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"802119874", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"119208478", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"12898848", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"959822", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"43733", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"895", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1036800"}], "+", RowBox[{"50108544", " ", SubscriptBox["\[Iota]", "5"]}], "-", RowBox[{"378482688", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "+", RowBox[{"1203879360", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "-", RowBox[{"1866851440", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "+", RowBox[{"901685472", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"1958467456", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"4787603188", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"5528964798", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"4152500203", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "+", RowBox[{"2141149820", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "-", RowBox[{"715311261", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "+", RowBox[{"98930622", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"42122506", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"32513904", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"11176602", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"2396078", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"327953", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"26348", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"935", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SubsuperscriptBox["\[Iota]", "4", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "7200"}], "+", RowBox[{"53760", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"130880", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"160564", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"115948", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"52522", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"15019", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"2548", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"202", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "4"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}]}], ")"}]}]}], ")"}]}]}]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "18"}], "n"], RowBox[{ FractionBox["1", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"], " ", SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "1"]}], ")"}], "3"]}]], RowBox[{ RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "1"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "1", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "1", "3"]}], "+", SubsuperscriptBox["\[Iota]", "1", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "16"}], SubscriptBox["\[Iota]", "1"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "2"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "2", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "2", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "2", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "2", "7"]}], "+", SubsuperscriptBox["\[Iota]", "2", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "3"], "=", "14"}], SubscriptBox["\[Iota]", "2"]], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "3"]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "+", SubsuperscriptBox["\[Iota]", "3", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "3", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "9"]}], "+", SubsuperscriptBox["\[Iota]", "3", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "4"], "=", "12"}], SubscriptBox["\[Iota]", "3"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", SubsuperscriptBox["\[Iota]", "4", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1036800"}], "+", RowBox[{"50108544", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"378482688", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"1203879360", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"1866851440", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"901685472", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "+", RowBox[{"1958467456", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "-", RowBox[{"4787603188", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "+", RowBox[{"5528964798", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"4152500203", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", RowBox[{"2141149820", " ", SubsuperscriptBox["\[Iota]", "4", "10"]}], "-", RowBox[{"715311261", " ", SubsuperscriptBox["\[Iota]", "4", "11"]}], "+", RowBox[{"98930622", " ", SubsuperscriptBox["\[Iota]", "4", "12"]}], "+", RowBox[{"42122506", " ", SubsuperscriptBox["\[Iota]", "4", "13"]}], "-", RowBox[{"32513904", " ", SubsuperscriptBox["\[Iota]", "4", "14"]}], "+", RowBox[{"11176602", " ", SubsuperscriptBox["\[Iota]", "4", "15"]}], "-", RowBox[{"2396078", " ", SubsuperscriptBox["\[Iota]", "4", "16"]}], "+", RowBox[{"327953", " ", SubsuperscriptBox["\[Iota]", "4", "17"]}], "-", RowBox[{"26348", " ", SubsuperscriptBox["\[Iota]", "4", "18"]}], "+", RowBox[{"935", " ", SubsuperscriptBox["\[Iota]", "4", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "20"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "5"], "=", "10"}], SubscriptBox["\[Iota]", "4"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "7200"}], "+", RowBox[{"53760", " ", SubscriptBox["\[Iota]", "5"]}], "-", RowBox[{"130880", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "+", RowBox[{"160564", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "-", RowBox[{"115948", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "+", RowBox[{"52522", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "-", RowBox[{"15019", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "+", RowBox[{"2548", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "-", RowBox[{"202", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "+", SubsuperscriptBox["\[Iota]", "5", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "268061736960"}], "+", RowBox[{"2824243780608", " ", SubscriptBox["\[Iota]", "5"]}], "+", RowBox[{"4318123156992", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "-", RowBox[{"201123626821632", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "+", RowBox[{"1391666715407232", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "-", RowBox[{"5277178048925952", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"12962808461111200", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"21305292915332800", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"21166111959705696", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"2890677091472172", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "-", RowBox[{"33448944651882246", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "+", RowBox[{"73993322390651126", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "-", RowBox[{"99697525001566438", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"100147841927725059", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"79546250838626709", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"50955754267106775", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"26260396787425091", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"10563977846009515", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"2989102604911489", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"315718564141377", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"242070039538587", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}], "-", RowBox[{"192965061962541", " ", SubsuperscriptBox["\[Iota]", "5", "21"]}], "+", RowBox[{"85146456468263", " ", SubsuperscriptBox["\[Iota]", "5", "22"]}], "-", RowBox[{"27671025884951", " ", SubsuperscriptBox["\[Iota]", "5", "23"]}], "+", RowBox[{"7076120002347", " ", SubsuperscriptBox["\[Iota]", "5", "24"]}], "-", RowBox[{"1452474125119", " ", SubsuperscriptBox["\[Iota]", "5", "25"]}], "+", RowBox[{"239656119375", " ", SubsuperscriptBox["\[Iota]", "5", "26"]}], "-", RowBox[{"31463978263", " ", SubsuperscriptBox["\[Iota]", "5", "27"]}], "+", RowBox[{"3217398027", " ", SubsuperscriptBox["\[Iota]", "5", "28"]}], "-", RowBox[{"247073926", " ", SubsuperscriptBox["\[Iota]", "5", "29"]}], "+", RowBox[{"13382758", " ", SubsuperscriptBox["\[Iota]", "5", "30"]}], "-", RowBox[{"452800", " ", SubsuperscriptBox["\[Iota]", "5", "31"]}], "+", RowBox[{"6936", " ", SubsuperscriptBox["\[Iota]", "5", "32"]}], "+", RowBox[{"16", " ", SubsuperscriptBox["\[Iota]", "5", "33"]}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "5"], "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "6"], "=", "8"}], SubscriptBox["\[Iota]", "5"]], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"14892318720", "-", RowBox[{"139725950976", " ", SubscriptBox["\[Iota]", "6"]}], "+", RowBox[{"544478712960", " ", SubsuperscriptBox["\[Iota]", "6", "2"]}], "-", RowBox[{"1213653772800", " ", SubsuperscriptBox["\[Iota]", "6", "3"]}], "+", RowBox[{"1752329541504", " ", SubsuperscriptBox["\[Iota]", "6", "4"]}], "-", RowBox[{"1727022580784", " ", SubsuperscriptBox["\[Iota]", "6", "5"]}], "+", RowBox[{"1160998559536", " ", SubsuperscriptBox["\[Iota]", "6", "6"]}], "-", RowBox[{"479387084584", " ", SubsuperscriptBox["\[Iota]", "6", "7"]}], "+", RowBox[{"42444457372", " ", SubsuperscriptBox["\[Iota]", "6", "8"]}], "+", RowBox[{"101229213030", " ", SubsuperscriptBox["\[Iota]", "6", "9"]}], "-", RowBox[{"88422557953", " ", SubsuperscriptBox["\[Iota]", "6", "10"]}], "+", RowBox[{"43907846563", " ", SubsuperscriptBox["\[Iota]", "6", "11"]}], "-", RowBox[{"15419001366", " ", SubsuperscriptBox["\[Iota]", "6", "12"]}], "+", RowBox[{"4044155236", " ", SubsuperscriptBox["\[Iota]", "6", "13"]}], "-", RowBox[{"802119874", " ", SubsuperscriptBox["\[Iota]", "6", "14"]}], "+", RowBox[{"119208478", " ", SubsuperscriptBox["\[Iota]", "6", "15"]}], "-", RowBox[{"12898848", " ", SubsuperscriptBox["\[Iota]", "6", "16"]}], "+", RowBox[{"959822", " ", SubsuperscriptBox["\[Iota]", "6", "17"]}], "-", RowBox[{"43733", " ", SubsuperscriptBox["\[Iota]", "6", "18"]}], "+", RowBox[{"895", " ", SubsuperscriptBox["\[Iota]", "6", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "6", "20"]}]}], ")"}], " ", RowBox[{"(", RowBox[{"593519176032819609600", "-", RowBox[{"10446329177297398333440", " ", SubscriptBox["\[Iota]", "6"]}], "+", RowBox[{"82630965219898476699648", " ", SubsuperscriptBox["\[Iota]", "6", "2"]}], "-", RowBox[{"357887593339887818373120", " ", SubsuperscriptBox["\[Iota]", "6", "3"]}], "+", RowBox[{"690983909070546888723456", " ", SubsuperscriptBox["\[Iota]", "6", "4"]}], "+", RowBox[{"1496932061461512796454400", " ", SubsuperscriptBox["\[Iota]", "6", "5"]}], "-", RowBox[{"16372134529175981041662720", " ", SubsuperscriptBox["\[Iota]", "6", "6"]}], "+", RowBox[{"66157720357287991244617728", " ", SubsuperscriptBox["\[Iota]", "6", "7"]}], "-", RowBox[{"176234670958207456090009472", " ", SubsuperscriptBox["\[Iota]", "6", "8"]}], "+", RowBox[{"340927853078980448736433216", " ", SubsuperscriptBox["\[Iota]", "6", "9"]}], "-", RowBox[{"478069445198356978374270464", " ", SubsuperscriptBox["\[Iota]", "6", "10"]}], "+", RowBox[{"422259594550509437294594320", " ", SubsuperscriptBox["\[Iota]", "6", "11"]}], "-", RowBox[{"1696454047051999626760968", " ", SubsuperscriptBox["\[Iota]", "6", "12"]}], "-", RowBox[{"827295585729170477467194152", " ", SubsuperscriptBox["\[Iota]", "6", "13"]}], "+", RowBox[{"1891616818293149574583751698", " ", SubsuperscriptBox["\[Iota]", "6", "14"]}], "-", RowBox[{"2852358842192182537015567718", " ", SubsuperscriptBox["\[Iota]", "6", "15"]}], "+", RowBox[{"3379568978845931753930388712", " ", SubsuperscriptBox["\[Iota]", "6", "16"]}], "-", RowBox[{"3327813430252425974324904640", " ", SubsuperscriptBox["\[Iota]", "6", "17"]}], "+", RowBox[{"2790700377405510701834118192", " ", SubsuperscriptBox["\[Iota]", "6", "18"]}], "-", RowBox[{"2014914026478703290951734807", " ", SubsuperscriptBox["\[Iota]", "6", "19"]}], "+", RowBox[{"1255359759286899924457407965", " ", SubsuperscriptBox["\[Iota]", "6", "20"]}], "-", RowBox[{"670558775247631998805980073", " ", SubsuperscriptBox["\[Iota]", "6", "21"]}], "+", RowBox[{"300843773637614981124313231", " ", SubsuperscriptBox["\[Iota]", "6", "22"]}], "-", RowBox[{"107316931132693335986435430", " ", SubsuperscriptBox["\[Iota]", "6", "23"]}], "+", RowBox[{"25083773398052438537473950", " ", SubsuperscriptBox["\[Iota]", "6", "24"]}], "+", RowBox[{"1155372404682645330065542", " ", SubsuperscriptBox["\[Iota]", "6", "25"]}], "-", RowBox[{"5437314367180455039158408", " ", SubsuperscriptBox["\[Iota]", "6", "26"]}], "+", RowBox[{"3807094879666162288177828", " ", SubsuperscriptBox["\[Iota]", "6", "27"]}], "-", RowBox[{"1865177834427973405930258", " ", SubsuperscriptBox["\[Iota]", "6", "28"]}], "+", RowBox[{"741345294491308012170232", " ", SubsuperscriptBox["\[Iota]", "6", "29"]}], "-", RowBox[{"250745206073700514078036", " ", SubsuperscriptBox["\[Iota]", "6", "30"]}], "+", RowBox[{"73677243614668746237912", " ", SubsuperscriptBox["\[Iota]", "6", "31"]}], "-", RowBox[{"18995086772703864765888", " ", SubsuperscriptBox["\[Iota]", "6", "32"]}], "+", RowBox[{"4316734897083386501414", " ", SubsuperscriptBox["\[Iota]", "6", "33"]}], "-", RowBox[{"865953430370743330500", " ", SubsuperscriptBox["\[Iota]", "6", "34"]}], "+", RowBox[{"153218767943719612259", " ", SubsuperscriptBox["\[Iota]", "6", "35"]}], "-", RowBox[{"23849092909382625867", " ", SubsuperscriptBox["\[Iota]", "6", "36"]}], "+", RowBox[{"3251585587952045721", " ", SubsuperscriptBox["\[Iota]", "6", "37"]}], "-", RowBox[{"385933937310432229", " ", SubsuperscriptBox["\[Iota]", "6", "38"]}], "+", RowBox[{"39548656479377428", " ", SubsuperscriptBox["\[Iota]", "6", "39"]}], "-", RowBox[{"3460902362671038", " ", SubsuperscriptBox["\[Iota]", "6", "40"]}], "+", RowBox[{"254885646365044", " ", SubsuperscriptBox["\[Iota]", "6", "41"]}], "-", RowBox[{"15487684236972", " ", SubsuperscriptBox["\[Iota]", "6", "42"]}], "+", RowBox[{"755062632096", " ", SubsuperscriptBox["\[Iota]", "6", "43"]}], "-", RowBox[{"28330156288", " ", SubsuperscriptBox["\[Iota]", "6", "44"]}], "+", RowBox[{"764221216", " ", SubsuperscriptBox["\[Iota]", "6", "45"]}], "-", RowBox[{"13001120", " ", SubsuperscriptBox["\[Iota]", "6", "46"]}], "+", RowBox[{"96704", " ", SubsuperscriptBox["\[Iota]", "6", "47"]}], "+", RowBox[{"256", " ", SubsuperscriptBox["\[Iota]", "6", "48"]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "+", SubscriptBox["\[Iota]", "6"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", SubscriptBox["\[Iota]", "6"]}], ")"}], " ", SubscriptBox["\[Iota]", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "412166094467235840"}], "+", RowBox[{"5717811264849119232", " ", SubscriptBox["\[Iota]", "6"]}], "-", RowBox[{"36138772924632064512", " ", SubsuperscriptBox["\[Iota]", "6", "2"]}], "+", RowBox[{"140349855651877949184", " ", SubsuperscriptBox["\[Iota]", "6", "3"]}], "-", RowBox[{"378777898422568498944", " ", SubsuperscriptBox["\[Iota]", "6", "4"]}], "+", RowBox[{"758763972872561806848", " ", SubsuperscriptBox["\[Iota]", "6", "5"]}], "-", RowBox[{"1172910092281534607296", " ", SubsuperscriptBox["\[Iota]", "6", "6"]}], "+", RowBox[{"1430909630074881248480", " ", SubsuperscriptBox["\[Iota]", "6", "7"]}], "-", RowBox[{"1390341094820656259632", " ", SubsuperscriptBox["\[Iota]", "6", "8"]}], "+", RowBox[{"1068467451890543337664", " ", SubsuperscriptBox["\[Iota]", "6", "9"]}], "-", RowBox[{"624704974810303880820", " ", SubsuperscriptBox["\[Iota]", "6", "10"]}], "+", RowBox[{"239348483335483302230", " ", SubsuperscriptBox["\[Iota]", "6", "11"]}], "-", RowBox[{"7040707094697754888", " ", SubsuperscriptBox["\[Iota]", "6", "12"]}], "-", RowBox[{"79932743247661774058", " ", SubsuperscriptBox["\[Iota]", "6", "13"]}], "+", RowBox[{"81112200716630610092", " ", SubsuperscriptBox["\[Iota]", "6", "14"]}], "-", RowBox[{"53282439119215791359", " ", SubsuperscriptBox["\[Iota]", "6", "15"]}], "+", RowBox[{"27238737541464309451", " ", SubsuperscriptBox["\[Iota]", "6", "16"]}], "-", RowBox[{"11465983444871365568", " ", SubsuperscriptBox["\[Iota]", "6", "17"]}], "+", RowBox[{"4072118966331964204", " ", SubsuperscriptBox["\[Iota]", "6", "18"]}], "-", RowBox[{"1234664080649927971", " ", SubsuperscriptBox["\[Iota]", "6", "19"]}], "+", RowBox[{"321362780236805789", " ", SubsuperscriptBox["\[Iota]", "6", "20"]}], "-", RowBox[{"71923499057928948", " ", SubsuperscriptBox["\[Iota]", "6", "21"]}], "+", RowBox[{"13823877306122378", " ", SubsuperscriptBox["\[Iota]", "6", "22"]}], "-", RowBox[{"2273119505706269", " ", SubsuperscriptBox["\[Iota]", "6", "23"]}], "+", RowBox[{"317735193578077", " ", SubsuperscriptBox["\[Iota]", "6", "24"]}], "-", RowBox[{"37399473131128", " ", SubsuperscriptBox["\[Iota]", "6", "25"]}], "+", RowBox[{"3658075291020", " ", SubsuperscriptBox["\[Iota]", "6", "26"]}], "-", RowBox[{"291823509303", " ", SubsuperscriptBox["\[Iota]", "6", "27"]}], "+", RowBox[{"18484998795", " ", SubsuperscriptBox["\[Iota]", "6", "28"]}], "-", RowBox[{"892856506", " ", SubsuperscriptBox["\[Iota]", "6", "29"]}], "+", RowBox[{"30772518", " ", SubsuperscriptBox["\[Iota]", "6", "30"]}], "-", RowBox[{"666304", " ", SubsuperscriptBox["\[Iota]", "6", "31"]}], "+", RowBox[{"6408", " ", SubsuperscriptBox["\[Iota]", "6", "32"]}], "+", RowBox[{"16", " ", SubsuperscriptBox["\[Iota]", "6", "33"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "268061736960"}], "+", RowBox[{"2824243780608", " ", SubscriptBox["\[Iota]", "6"]}], "+", RowBox[{"4318123156992", " ", SubsuperscriptBox["\[Iota]", "6", "2"]}], "-", RowBox[{"201123626821632", " ", SubsuperscriptBox["\[Iota]", "6", "3"]}], "+", RowBox[{"1391666715407232", " ", SubsuperscriptBox["\[Iota]", "6", "4"]}], "-", RowBox[{"5277178048925952", " ", SubsuperscriptBox["\[Iota]", "6", "5"]}], "+", RowBox[{"12962808461111200", " ", SubsuperscriptBox["\[Iota]", "6", "6"]}], "-", RowBox[{"21305292915332800", " ", SubsuperscriptBox["\[Iota]", "6", "7"]}], "+", RowBox[{"21166111959705696", " ", SubsuperscriptBox["\[Iota]", "6", "8"]}], "-", RowBox[{"2890677091472172", " ", SubsuperscriptBox["\[Iota]", "6", "9"]}], "-", RowBox[{"33448944651882246", " ", SubsuperscriptBox["\[Iota]", "6", "10"]}], "+", RowBox[{"73993322390651126", " ", SubsuperscriptBox["\[Iota]", "6", "11"]}], "-", RowBox[{"99697525001566438", " ", SubsuperscriptBox["\[Iota]", "6", "12"]}], "+", RowBox[{"100147841927725059", " ", SubsuperscriptBox["\[Iota]", "6", "13"]}], "-", RowBox[{"79546250838626709", " ", SubsuperscriptBox["\[Iota]", "6", "14"]}], "+", RowBox[{"50955754267106775", " ", SubsuperscriptBox["\[Iota]", "6", "15"]}], "-", RowBox[{"26260396787425091", " ", SubsuperscriptBox["\[Iota]", "6", "16"]}], "+", RowBox[{"10563977846009515", " ", SubsuperscriptBox["\[Iota]", "6", "17"]}], "-", RowBox[{"2989102604911489", " ", SubsuperscriptBox["\[Iota]", "6", "18"]}], "+", RowBox[{"315718564141377", " ", SubsuperscriptBox["\[Iota]", "6", "19"]}], "+", RowBox[{"242070039538587", " ", SubsuperscriptBox["\[Iota]", "6", "20"]}], "-", RowBox[{"192965061962541", " ", SubsuperscriptBox["\[Iota]", "6", "21"]}], "+", RowBox[{"85146456468263", " ", SubsuperscriptBox["\[Iota]", "6", "22"]}], "-", RowBox[{"27671025884951", " ", SubsuperscriptBox["\[Iota]", "6", "23"]}], "+", RowBox[{"7076120002347", " ", SubsuperscriptBox["\[Iota]", "6", "24"]}], "-", RowBox[{"1452474125119", " ", SubsuperscriptBox["\[Iota]", "6", "25"]}], "+", RowBox[{"239656119375", " ", SubsuperscriptBox["\[Iota]", "6", "26"]}], "-", RowBox[{"31463978263", " ", SubsuperscriptBox["\[Iota]", "6", "27"]}], "+", RowBox[{"3217398027", " ", SubsuperscriptBox["\[Iota]", "6", "28"]}], "-", RowBox[{"247073926", " ", SubsuperscriptBox["\[Iota]", "6", "29"]}], "+", RowBox[{"13382758", " ", SubsuperscriptBox["\[Iota]", "6", "30"]}], "-", RowBox[{"452800", " ", SubsuperscriptBox["\[Iota]", "6", "31"]}], "+", RowBox[{"6936", " ", SubsuperscriptBox["\[Iota]", "6", "32"]}], "+", RowBox[{"16", " ", SubsuperscriptBox["\[Iota]", "6", "33"]}]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"14892318720", "-", RowBox[{"139725950976", " ", SubscriptBox["\[Iota]", "5"]}], "+", RowBox[{"544478712960", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "-", RowBox[{"1213653772800", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "+", RowBox[{"1752329541504", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "-", RowBox[{"1727022580784", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"1160998559536", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"479387084584", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"42444457372", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "+", RowBox[{"101229213030", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "-", RowBox[{"88422557953", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "+", RowBox[{"43907846563", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "-", RowBox[{"15419001366", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"4044155236", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"802119874", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"119208478", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"12898848", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"959822", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"43733", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"895", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1036800"}], "+", RowBox[{"50108544", " ", SubscriptBox["\[Iota]", "5"]}], "-", RowBox[{"378482688", " ", SubsuperscriptBox["\[Iota]", "5", "2"]}], "+", RowBox[{"1203879360", " ", SubsuperscriptBox["\[Iota]", "5", "3"]}], "-", RowBox[{"1866851440", " ", SubsuperscriptBox["\[Iota]", "5", "4"]}], "+", RowBox[{"901685472", " ", SubsuperscriptBox["\[Iota]", "5", "5"]}], "+", RowBox[{"1958467456", " ", SubsuperscriptBox["\[Iota]", "5", "6"]}], "-", RowBox[{"4787603188", " ", SubsuperscriptBox["\[Iota]", "5", "7"]}], "+", RowBox[{"5528964798", " ", SubsuperscriptBox["\[Iota]", "5", "8"]}], "-", RowBox[{"4152500203", " ", SubsuperscriptBox["\[Iota]", "5", "9"]}], "+", RowBox[{"2141149820", " ", SubsuperscriptBox["\[Iota]", "5", "10"]}], "-", RowBox[{"715311261", " ", SubsuperscriptBox["\[Iota]", "5", "11"]}], "+", RowBox[{"98930622", " ", SubsuperscriptBox["\[Iota]", "5", "12"]}], "+", RowBox[{"42122506", " ", SubsuperscriptBox["\[Iota]", "5", "13"]}], "-", RowBox[{"32513904", " ", SubsuperscriptBox["\[Iota]", "5", "14"]}], "+", RowBox[{"11176602", " ", SubsuperscriptBox["\[Iota]", "5", "15"]}], "-", RowBox[{"2396078", " ", SubsuperscriptBox["\[Iota]", "5", "16"]}], "+", RowBox[{"327953", " ", SubsuperscriptBox["\[Iota]", "5", "17"]}], "-", RowBox[{"26348", " ", SubsuperscriptBox["\[Iota]", "5", "18"]}], "+", RowBox[{"935", " ", SubsuperscriptBox["\[Iota]", "5", "19"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "5", "20"]}]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SubscriptBox["\[Iota]", "4"]}], ")"}], "3"], " ", SubsuperscriptBox["\[Iota]", "4", "2"], " ", RowBox[{"(", RowBox[{"1", "+", SubscriptBox["\[Iota]", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "7200"}], "+", RowBox[{"53760", " ", SubscriptBox["\[Iota]", "4"]}], "-", RowBox[{"130880", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "+", RowBox[{"160564", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"115948", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"52522", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"15019", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"2548", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"202", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "-", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}], " ", RowBox[{"(", RowBox[{"144", "-", RowBox[{"1392", " ", SubscriptBox["\[Iota]", "4"]}], "+", RowBox[{"2884", " ", SubsuperscriptBox["\[Iota]", "4", "2"]}], "-", RowBox[{"568", " ", SubsuperscriptBox["\[Iota]", "4", "3"]}], "-", RowBox[{"3625", " ", SubsuperscriptBox["\[Iota]", "4", "4"]}], "+", RowBox[{"4604", " ", SubsuperscriptBox["\[Iota]", "4", "5"]}], "-", RowBox[{"2797", " ", SubsuperscriptBox["\[Iota]", "4", "6"]}], "+", RowBox[{"980", " ", SubsuperscriptBox["\[Iota]", "4", "7"]}], "-", RowBox[{"175", " ", SubsuperscriptBox["\[Iota]", "4", "8"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "4", "9"]}], "+", SubsuperscriptBox["\[Iota]", "4", "10"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "192"}], "+", RowBox[{"1396", " ", SubscriptBox["\[Iota]", "3"]}], "-", RowBox[{"3038", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"3165", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"1818", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"579", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"85", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}], " ", RowBox[{"(", RowBox[{"8", "-", RowBox[{"64", " ", SubscriptBox["\[Iota]", "3"]}], "+", RowBox[{"92", " ", SubsuperscriptBox["\[Iota]", "3", "2"]}], "+", RowBox[{"39", " ", SubsuperscriptBox["\[Iota]", "3", "3"]}], "-", RowBox[{"128", " ", SubsuperscriptBox["\[Iota]", "3", "4"]}], "+", RowBox[{"125", " ", SubsuperscriptBox["\[Iota]", "3", "5"]}], "-", RowBox[{"57", " ", SubsuperscriptBox["\[Iota]", "3", "6"]}], "+", RowBox[{"8", " ", SubsuperscriptBox["\[Iota]", "3", "7"]}], "+", SubsuperscriptBox["\[Iota]", "3", "8"]}], ")"}]}], ")"}]}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"22", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"15", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"2", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}], " ", RowBox[{"(", RowBox[{"6", "+", RowBox[{"2", " ", SubscriptBox["\[Iota]", "2"]}], "-", RowBox[{"3", " ", SubsuperscriptBox["\[Iota]", "2", "2"]}], "+", RowBox[{"6", " ", SubsuperscriptBox["\[Iota]", "2", "3"]}], "+", SubsuperscriptBox["\[Iota]", "2", "4"]}], ")"}]}], ")"}]}]}], ")"}]}]}]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.949641123723929*^9}, CellLabel->"Out[18]=",ExpressionUUID->"c356f423-a1b0-4725-9408-4c1968e57f1e"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recSol", "=", RowBox[{"SolveRecurrence", "[", RowBox[{"rec", ",", RowBox[{"f", "[", "n", "]"}], ",", RowBox[{"SigmaPrint", "\[Rule]", "True"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.949317259877136*^9, 3.94931726125134*^9}}, CellLabel->"In[19]:=",ExpressionUUID->"3509eb43-f0b8-4fb8-aeb4-d2ee362f7e06"], Cell[CellGroupData[{ Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "7"}], SequenceForm["Solve recurrence of order: ", 7], Editable->False]], "Print", CellChangeTimes->{3.9496411240953903`*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"289d926e-5db1-43b1-9ded-c4b2833444ba"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "6"}], SequenceForm["Solve recurrence of order: ", 6], Editable->False]], "Print", CellChangeTimes->{3.949641125115553*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"a900a013-db5a-44bd-a0f0-fa72e4db7c43"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "5"}], SequenceForm["Solve recurrence of order: ", 5], Editable->False]], "Print", CellChangeTimes->{3.949641126382855*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"ed010ce0-1456-4daa-86e3-00da3eac20e2"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "4"}], SequenceForm["Solve recurrence of order: ", 4], Editable->False]], "Print", CellChangeTimes->{3.94964112648272*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"bc51e80e-8163-401a-bb09-491409a3ca58"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "3"}], SequenceForm["Solve recurrence of order: ", 3], Editable->False]], "Print", CellChangeTimes->{3.949641126622656*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"e1116f1f-1338-4074-baa6-59c7ae25c2f9"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "2"}], SequenceForm["Solve recurrence of order: ", 2], Editable->False]], "Print", CellChangeTimes->{3.949641128047804*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"c86bc5fe-535f-4508-b525-a72ec7840934"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "1"}], SequenceForm["Solve recurrence of order: ", 1], Editable->False]], "Print", CellChangeTimes->{3.949641128730277*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"93154286-0e62-4d5c-b643-c0be9385b86f"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Solve recurrence of order: \"\>", "\[InvisibleSpace]", "0"}], SequenceForm["Solve recurrence of order: ", 0], Editable->False]], "Print", CellChangeTimes->{3.949641130624254*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"fdd77f12-207d-4aa6-81c9-56e6f8cf00b3"], Cell[BoxData["\<\"Simplification of solutions.\"\>"], "Print", CellChangeTimes->{3.949641130645977*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"be3fd421-9bc3-487d-9363-b2f0487a34f4"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Total number of extensions and the total amount of bytes: \ [\"\>", "\[InvisibleSpace]", "19", "\[InvisibleSpace]", "\<\",\"\>", "\[InvisibleSpace]", "144552", "\[InvisibleSpace]", "\<\"]\"\>"}], SequenceForm[ "Total number of extensions and the total amount of bytes: [", 19, ",", 144552, "]"], Editable->False]], "Print", CellChangeTimes->{3.949641130919276*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"225e76bf-bcfa-4e0d-99f0-08876fd6e5c9"], Cell[BoxData["\<\"Current extension and its bytesize; currently used \ extensions:\"\>"], "Print", CellChangeTimes->{3.949641130924024*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"71e31919-22a0-4d14-a588-f86ffe5519d8"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"1", ",", RowBox[{"160", ";", "1"}]}], "]"}], " ", "[", RowBox[{"2", ",", RowBox[{"1624", ";", "2"}]}], "]"}], " ", "[", RowBox[{"3", ",", RowBox[{"1840", ";", "3"}]}], "]"}], " ", "[", RowBox[{"4", ",", RowBox[{"2056", ";", "4"}]}], "]"}], " ", "[", RowBox[{"5", ",", RowBox[{"11272", ";", "5"}]}], "]"}], " ", "[", RowBox[{"6", ",", RowBox[{"13760", ";", "5"}]}], "]"}], " ", "[", RowBox[{"7", ",", RowBox[{"12272", ";", "5"}]}], "]"}], " ", "[", RowBox[{"8", ",", RowBox[{"5960", ";", "5"}]}], "]"}], " "}], "\n", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"9", ",", RowBox[{"18264", ";", "7"}]}], "]"}], " ", "[", RowBox[{"10", ",", RowBox[{"11624", ";", "7"}]}], "]"}], " ", "[", RowBox[{"11", ",", RowBox[{"13544", ";", "8"}]}], "]"}], " ", "[", RowBox[{"12", ",", RowBox[{"11888", ";", "8"}]}], "]"}], " ", "[", RowBox[{"13", ",", RowBox[{"5360", ";", "8"}]}], "]"}], " ", "[", RowBox[{"14", ",", RowBox[{"31544", ";", "10"}]}], "]"}], " ", "[", RowBox[{"15", ",", RowBox[{"34840", ";", "10"}]}], "]"}], " "}], "\n", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"16", ",", RowBox[{"31720", ";", "10"}]}], "]"}], " ", "[", RowBox[{"17", ",", RowBox[{"21496", ";", "10"}]}], "]"}], " ", "[", RowBox[{"18", ",", RowBox[{"28400", ";", "10"}]}], "]"}], " ", "[", RowBox[{"19", ",", RowBox[{"13400", ";", "10"}]}], "]"}], " "}]}], "Print", CellTags-> "7899289383",ExpressionUUID->"7cd71215-6315-4c1f-a2b6-41d64fa13134"], Cell[BoxData["\<\"Computation of lower bounds.\"\>"], "Print", CellChangeTimes->{3.949641150848034*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"f02c40e5-ba97-4c39-a476-c883b5518c0a"], Cell[BoxData["\<\"Interpretation of solutions as sums and products.\"\>"], \ "Print", CellChangeTimes->{3.949641150889485*^9}, CellLabel-> "During evaluation of \ In[19]:=",ExpressionUUID->"b4dcf747-5d79-4ff3-9d40-9cff55508f82"] }, Open ]], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ FractionBox[ RowBox[{"3", "+", RowBox[{"6", " ", "n"}], "+", RowBox[{"6", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["n", "3"]}]}], RowBox[{"2", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ FractionBox[ RowBox[{"198", "+", RowBox[{"382", " ", "n"}], "+", RowBox[{"383", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"294", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"29", " ", SuperscriptBox["n", "4"]}]}], RowBox[{"2", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", RowBox[{"66", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "+", RowBox[{"14", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ FractionBox[ RowBox[{"33", "+", RowBox[{"65", " ", "n"}], "+", RowBox[{"65", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"46", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"2", " ", SuperscriptBox["n", "4"]}]}], RowBox[{"2", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", RowBox[{"11", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]]}], "-", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "-", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ FractionBox["2", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]]}], ")"}]}], "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", " ", "n"}], "+", RowBox[{"2", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], RowBox[{"2", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"]}]], "+", RowBox[{"4", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", RowBox[{"2", " ", "n"}], "-", RowBox[{"2", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], RowBox[{"n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}], ")"}]}], "-", RowBox[{"4", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{ FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "71"}], "-", RowBox[{"220", " ", "n"}], "-", RowBox[{"308", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"260", " ", SuperscriptBox["n", "3"]}], "-", RowBox[{"98", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"15", " ", SuperscriptBox["n", "5"]}]}], ")"}]}], RowBox[{"3", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]}]], "+", RowBox[{"10", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"20", " ", "n"}], "+", RowBox[{"20", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], RowBox[{"n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"3", "+", RowBox[{"6", " ", "n"}], "+", RowBox[{"6", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"4", " ", SuperscriptBox["n", "3"]}]}], RowBox[{ SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "+", RowBox[{"6", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{"8", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.949641151183393*^9}, CellLabel->"Out[19]=",ExpressionUUID->"19e45111-bcf3-4557-91f6-486b3f8eee9f"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"FindLinearCombination", "[", RowBox[{"recSol", ",", RowBox[{"{", RowBox[{"3", ",", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["63653", "3888"]}], ",", RowBox[{"-", FractionBox["8802581", "486000"]}], ",", RowBox[{"-", FractionBox["5234029", "270000"]}], ",", RowBox[{"-", FractionBox["13978373", "686000"]}], ",", RowBox[{"-", FractionBox["15827913259", "746807040"]}], ",", RowBox[{"-", FractionBox["1578915745223", "72013536000"]}], ",", RowBox[{"-", FractionBox["95194737209", "4219543125"]}]}], "}"}]}], "}"}], ",", "n", ",", "7"}], "]"}]}]], "Input", CellChangeTimes->{3.550749229759121*^9}, CellLabel->"In[20]:=",ExpressionUUID->"5f8a7beb-04dc-43b6-a98d-60c3af2fb8fc"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "132"}], "+", RowBox[{"106", " ", "n"}], "+", RowBox[{"507", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"571", " ", SuperscriptBox["n", "3"]}], "-", RowBox[{"979", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"383", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"1080", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"270", " ", SuperscriptBox["n", "7"]}]}], ")"}]}], RowBox[{"27", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]}]]}], "+", FractionBox[ RowBox[{"64", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], RowBox[{"9", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]}]], "+", RowBox[{ FractionBox["80", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], "-", FractionBox[ RowBox[{"8", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"14", " ", "n"}], "+", RowBox[{"14", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], RowBox[{"3", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ FractionBox["1336", "27"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "-", FractionBox[ RowBox[{"4", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "54"}], "-", RowBox[{"108", " ", "n"}], "-", RowBox[{"108", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"137", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"627", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"627", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"209", " ", SuperscriptBox["n", "6"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"27", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", FractionBox[ RowBox[{"32", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"3", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "+", RowBox[{"16", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{ FractionBox["64", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], "+", FractionBox[ RowBox[{"16", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "9"}], "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"10", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], RowBox[{"9", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{"32", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "+", FractionBox[ RowBox[{"16", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"16", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], RowBox[{"9", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"]}]], "-", FractionBox[ RowBox[{"32", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], RowBox[{"3", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "-", RowBox[{ FractionBox["64", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "-", RowBox[{ FractionBox["128", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}]}], "-", RowBox[{ FractionBox["32", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "-", FractionBox[ RowBox[{"32", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"10", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{"9", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "-", RowBox[{ FractionBox["64", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{ FractionBox["128", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], SubsuperscriptBox["\[Iota]", "2", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}]], "Output", CellChangeTimes->{3.949641151577552*^9}, CellLabel->"Out[20]=",ExpressionUUID->"940a00d4-1a77-423e-9223-117532f72c96"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "=", RowBox[{ RowBox[{ RowBox[{"sol", "//", "TransformToSSums"}], "//", "ReduceToBasis"}], "//", "CollectProdSum"}]}]], "Input", CellChangeTimes->{{3.87847427339078*^9, 3.8784742839305763`*^9}, { 3.878474497770916*^9, 3.8784744989694853`*^9}, {3.94931730283497*^9, 3.949317309482458*^9}}, CellLabel->"In[21]:=",ExpressionUUID->"3b5cebe6-6cd3-4870-9c84-b9641d5d439f"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "132"}], "+", RowBox[{"106", " ", "n"}], "+", RowBox[{"507", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"571", " ", SuperscriptBox["n", "3"]}], "-", RowBox[{"979", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"383", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"1080", " ", SuperscriptBox["n", "6"]}], "+", RowBox[{"270", " ", SuperscriptBox["n", "7"]}]}], ")"}]}], RowBox[{"27", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]}]]}], "+", FractionBox[ RowBox[{"64", " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], RowBox[{"9", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "4"]}]], "+", RowBox[{ FractionBox["32", "3"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "4"}], ",", "n"}], "]"}]}], "+", FractionBox[ RowBox[{"16", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"16", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "n"}], "]"}]}], RowBox[{"9", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"]}]], "+", RowBox[{ RowBox[{"S", "[", RowBox[{ RowBox[{"-", "3"}], ",", "n"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"16", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"10", " ", SuperscriptBox["n", "2"]}]}], ")"}]}], RowBox[{"9", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]]}], "+", RowBox[{ FractionBox["32", "3"], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}]}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ FractionBox["1336", "27"], "+", RowBox[{ FractionBox["32", "3"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "n"}], "]"}]}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{"2", ",", "n"}], "]"}]}], "-", FractionBox[ RowBox[{"8", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"14", " ", "n"}], "+", RowBox[{"14", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{"3", ",", "n"}], "]"}]}], RowBox[{"3", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ FractionBox["80", "3"], " ", RowBox[{"S", "[", RowBox[{"4", ",", "n"}], "]"}]}], "+", FractionBox[ RowBox[{"32", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "3"}], "+", RowBox[{"10", " ", "n"}], "+", RowBox[{"10", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "n"}], "]"}]}], RowBox[{"9", " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}]}]], "+", RowBox[{ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"4", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "54"}], "-", RowBox[{"108", " ", "n"}], "-", RowBox[{"108", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"137", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"627", " ", SuperscriptBox["n", "4"]}], "+", RowBox[{"627", " ", SuperscriptBox["n", "5"]}], "+", RowBox[{"209", " ", SuperscriptBox["n", "6"]}]}], ")"}]}], RowBox[{"27", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]]}], "-", FractionBox[ RowBox[{"32", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], RowBox[{"3", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]}]], "-", RowBox[{ FractionBox["320", "9"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "n"}], "]"}]}], "+", RowBox[{"16", " ", RowBox[{"S", "[", RowBox[{"3", ",", "n"}], "]"}]}], "+", RowBox[{ FractionBox["64", "3"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "n"}], "]"}]}]}], ")"}]}], "+", RowBox[{ FractionBox["32", "3"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "2", ",", "n"}], "]"}]}], "-", RowBox[{ FractionBox["64", "3"], " ", RowBox[{"S", "[", RowBox[{"3", ",", "1", ",", "n"}], "]"}]}], "-", RowBox[{ FractionBox["128", "3"], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "1", ",", "n"}], "]"}]}]}]], "Output",\ CellChangeTimes->{3.9496411519877443`*^9}, CellLabel->"Out[21]=",ExpressionUUID->"d6742d78-638b-4ada-8060-9374367ea7bc"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Automatic evaluation of sums", "Section", CellChangeTimes->{{3.695359119957041*^9, 3.695359122800109*^9}, { 3.9493208310997467`*^9, 3.949320833390876*^9}},ExpressionUUID->"88303882-1f2f-4f52-a441-\ 9f31c1527641"], Cell[CellGroupData[{ Cell["The physics example", "Subsection", CellChangeTimes->{{3.9493208021602592`*^9, 3.949320809405171*^9}},ExpressionUUID->"a72b17d1-ab44-4073-a4ea-\ 96381a4aafe4"], Cell[BoxData[ RowBox[{ RowBox[{"summand", "=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"N", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "N"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "N"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "N"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}]}], ")"}]}]}], ";"}]], "Input", CellChangeTimes->{{3.695358996200685*^9, 3.695358999573455*^9}}, CellLabel->"In[22]:=",ExpressionUUID->"6e2a3ddc-96ea-4346-a9f3-db9a18c3b252"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "\[Infinity]"], "summand"}]}]}]], "Input", CellChangeTimes->{{3.695359054949247*^9, 3.695359058352847*^9}}, CellLabel->"In[23]:=",ExpressionUUID->"c783cbdd-bb52-4984-985d-8ad7887415bb"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "\[Infinity]"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"j", SubscriptBox["!", "."]}], " ", RowBox[{"k", SubscriptBox["!", "."]}], " ", RowBox[{"N", SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"j", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"2", "+", RowBox[{"2", " ", "j"}], "+", "k", "+", "N"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}]}]], "-", RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k"}]}], "]"}], "+", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "N"}]}], "]"}], "-", RowBox[{"S", "[", RowBox[{"1", ",", RowBox[{"j", "+", "k", "+", "N"}]}], "]"}]}], ")"}]}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "k"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "j", "+", "N"}], ")"}], SubscriptBox["!", "."]}], " ", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "N"}], ")"}], SubscriptBox["!", "."]}]}], ")"}]}]}]}]], "Output", CellChangeTimes->{3.949641152022346*^9}, CellLabel->"Out[23]=",ExpressionUUID->"68d73dc8-1d95-4e78-9da7-8eab59d7f9e1"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EvaluateMultiSum", "[", RowBox[{"mySum", ",", RowBox[{"{", "}"}], ",", RowBox[{"{", "N", "}"}], ",", RowBox[{"{", "1", "}"}], ",", RowBox[{"{", "Infinity", "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.695359067258122*^9, 3.695359084209755*^9}}, CellLabel->"In[24]:=",ExpressionUUID->"e3e91021-4933-402f-9b5f-e1c8ce4ca7f8"], Cell[BoxData[ RowBox[{ FractionBox[ SuperscriptBox[ RowBox[{"S", "[", RowBox[{"1", ",", "N"}], "]"}], "2"], RowBox[{"2", " ", "N", " ", RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}]}]], "+", FractionBox[ RowBox[{"3", " ", RowBox[{"S", "[", RowBox[{"2", ",", "N"}], "]"}]}], RowBox[{"2", " ", "N", " ", RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}]}]]}]], "Output", CellChangeTimes->{3.9496411741107597`*^9}, CellLabel->"Out[24]=",ExpressionUUID->"0f76f322-bdd4-4570-810f-d128583f6e17"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["0.999 ... = 1?", "Subsection", CellChangeTimes->{{3.9493175220329123`*^9, 3.949317542955522*^9}},ExpressionUUID->"11c1b408-c5d8-46a7-9e2a-\ f89c68d1a94b"], Cell[BoxData[ RowBox[{ RowBox[{"mySum", " ", "=", " ", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "\[Infinity]"], FractionBox[ RowBox[{ RowBox[{"S", "[", RowBox[{"1", ",", "j"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", FractionBox["1", RowBox[{"1", "+", "k"}]], "+", RowBox[{"S", "[", RowBox[{"1", ",", "k"}], "]"}]}], ")"}]}], RowBox[{"j", " ", "k", " ", RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"j", "+", "k"}], ")"}]}]]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.9493175220329123`*^9, 3.949317546505063*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"fe7f6a19-52e8-4395-87df-8c97f04f3bff"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EvaluateMultiSum", "[", RowBox[{"mySum", ",", " ", RowBox[{"{", "}"}], ",", " ", RowBox[{"{", "}"}], ",", " ", RowBox[{"{", "}"}], ",", " ", RowBox[{"{", "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.9493175220329123`*^9, 3.949317546505063*^9}}, CellLabel->"In[26]:=",ExpressionUUID->"28f72394-a9a2-446f-8da8-71bc889d64cf"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", "z3"}], "+", RowBox[{"z2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", RowBox[{"4", " ", "z3"}]}], ")"}]}], "+", RowBox[{"2", " ", "z5"}]}]], "Output", CellChangeTimes->{3.949641181125952*^9}, CellLabel->"Out[26]=",ExpressionUUID->"5b5e71e2-aed2-4bee-8a2b-089548a8d02e"] }, Open ]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part 2", "Title", CellChangeTimes->{{3.9493180982012577`*^9, 3.949318102279457*^9}},ExpressionUUID->"257c97cb-0133-4b0c-ad37-\ ed68ef03eb21"], Cell[CellGroupData[{ Cell["Refined telescoping", "Subsection", CellChangeTimes->{{3.9494179253969603`*^9, 3.9494179282777987`*^9}},ExpressionUUID->"b8b85520-fee3-4236-9f2d-\ 3be78b5a71dc"], Cell[BoxData[ RowBox[{ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "6"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "5"], "=", "1"}], SubscriptBox["\[Omicron]", "6"]], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "4"], "=", "1"}], SubscriptBox["\[Omicron]", "5"]], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "3"], "=", "1"}], SubscriptBox["\[Omicron]", "4"]], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], SubscriptBox["\[Omicron]", "3"]], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], SubscriptBox["\[Omicron]", "3"]]}], SubscriptBox["\[Omicron]", "4"]]}], SubscriptBox["\[Omicron]", "5"]]}], SubscriptBox["\[Omicron]", "6"]]}]}], ";"}]], "Input", CellChangeTimes->{{3.949417979338838*^9, 3.949417992494116*^9}}, CellLabel->"In[27]:=",ExpressionUUID->"d5fa5f4d-5bd5-44b1-920f-548dd46257dc"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.949417965466279*^9, 3.949417985007909*^9}}, CellLabel->"In[28]:=",ExpressionUUID->"af9459e8-488b-4b89-a56f-399d2ca97d1b"], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "5"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "8"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "2"]}], "+", RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "3"]}], "+", RowBox[{ FractionBox["1", "120"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "5"]}]}]], "Output", CellChangeTimes->{3.949641192816259*^9}, CellLabel->"Out[28]=",ExpressionUUID->"56ed1b9a-42d1-4a27-8230-b0a5b52cf3c3"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ",", RowBox[{"Tower", "->", RowBox[{"{", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "3"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], SubscriptBox["\[Omicron]", "3"]], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], SubscriptBox["\[Omicron]", "3"]]}], "}"}]}], ",", RowBox[{"SimplifyByExt", "->", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.9494180876467648`*^9, 3.949418169349887*^9}, { 3.9494182004244843`*^9, 3.949418203248124*^9}}, CellLabel->"In[29]:=",ExpressionUUID->"c61d623f-db9c-4250-8c2b-a020c0c4e128"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], "2"]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}]], "Output", CellChangeTimes->{3.949641193189834*^9}, CellLabel->"Out[29]=",ExpressionUUID->"51360d63-2fa8-4d92-beeb-e024c8e3ffbc"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "4"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "3"], "=", "1"}], SubscriptBox["\[Omicron]", "4"]], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], SubscriptBox["\[Omicron]", "3"]], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], SubsuperscriptBox["\[Omicron]", "3", "2"]]}], SubsuperscriptBox["\[Omicron]", "4", "2"]]}]}], ";"}]], "Input", CellChangeTimes->{{3.949418050502849*^9, 3.9494180556236486`*^9}}, CellLabel->"In[30]:=",ExpressionUUID->"e5dee354-661a-42c7-9b14-febbb13829ce"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.949418002095335*^9, 3.9494180228492393`*^9}, { 3.949418064214164*^9, 3.9494180652272873`*^9}}, CellLabel->"In[31]:=",ExpressionUUID->"960c8fb2-fe7d-4301-b7eb-88f6080d5dff"], Cell[BoxData[ RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "4"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "2"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}]}]], "Output", CellChangeTimes->{3.949641193834996*^9}, CellLabel->"Out[31]=",ExpressionUUID->"0de78a72-735e-4cb4-aa13-7e643d602bd4"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"Calkin2", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "0"}], "k"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"a"}, {"i"} }], ")"}], "."]}], ")"}], "2"]}]}], ";"}]], "Input", CellLabel->"In[32]:=",ExpressionUUID->"f1a86c25-c098-441b-996a-1d303cf49943"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"res", "=", RowBox[{"SigmaReduce", "[", "Calkin2", "]"}]}]], "Input", CellChangeTimes->{3.949459032065844*^9}, CellLabel->"In[33]:=",ExpressionUUID->"cfc26608-d196-461b-9387-0fa737f711f3"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"a", "-", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"a"}, {"n"} }], ")"}], "."], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"a"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."]}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"2", "-", "a", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], SuperscriptBox[ RowBox[{"(", GridBox[{ {"a"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."]}], ")"}], "2"]}], "-", RowBox[{ FractionBox["1", "2"], " ", "a", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], SuperscriptBox[ RowBox[{"(", SuperscriptBox[ RowBox[{"(", GridBox[{ {"a"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], ")"}], "2"]}], ")"}]}]}]], "Output", CellChangeTimes->{3.9496411949875298`*^9}, CellLabel->"Out[33]=",ExpressionUUID->"6f007631-7747-4aec-84bf-fab37c1c6275"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"qSum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], GridBox[{ { RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}]} }], "N"], RowBox[{"(", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]], " ", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], GridBox[{ { RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}]} }], SubscriptBox["\[Iota]", "1"]], RowBox[{"(", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]], " ", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], GridBox[{ { RowBox[{ SubscriptBox["\[Iota]", "3"], "=", "1"}]} }], SubscriptBox["\[Iota]", "2"]], RowBox[{"(", FractionBox[ SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "3"], "."]], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "3"], "."]]}]}]], ")"}]}]}], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "2"], "."]]}]}]], ")"}]}]}], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox[ RowBox[{"(", "q", ")"}], UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}]], ")"}]}]}], ";"}]], "Input",\ CellChangeTimes->{{3.949464667971999*^9, 3.949464668680031*^9}}, CellLabel->"In[34]:=",ExpressionUUID->"d41dd618-a09c-4512-a5d5-4a674f3f4f13"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", "qSum", "]"}]], "Input", CellChangeTimes->{ 3.949459098623344*^9, {3.949464673628998*^9, 3.9494646771248083`*^9}}, CellLabel->"In[35]:=",ExpressionUUID->"cf18441e-9caf-4ef3-a334-4bc57e40444d"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"N", " ", RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "N"}], ")"}]}], RowBox[{"6", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}], ")"}], "3"]]}], RowBox[{"3", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"2", "+", "N"}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}], ")"}], "2"]]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "N"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "N"}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}]]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}], ")"}], "2"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}]]}], ")"}]}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"2", "+", "N"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}]]}], ")"}], "2"]}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "N"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{ SuperscriptBox["q", UnderscriptBox["K", "."]], " ", SuperscriptBox["q", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}]]}], ")"}], "3"], RowBox[{"6", " ", SuperscriptBox[ RowBox[{"(", SuperscriptBox["q", UnderscriptBox["K", "."]], ")"}], "3"]}]]}]], "Output", CellChangeTimes->{3.9496411972130823`*^9}, CellLabel->"Out[35]=",ExpressionUUID->"21da9b8c-beed-45c8-b353-ac6fd1144308"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "K"], FractionBox[ RowBox[{"x", "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], UnderscriptBox[ RowBox[{"1", "+", "m"}], "."]], " ", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "k"], FractionBox[ RowBox[{ SubscriptBox["\[Iota]", "1"], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"m", "+", SubscriptBox["\[Iota]", "1"]}]}, {"m"} }], ")"}], "."], " ", SuperscriptBox["x", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}], RowBox[{"m", "+", SubscriptBox["\[Iota]", "1"]}]]}]}]}], RowBox[{"x", " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"k", "+", "m"}]}, {"n"} }], ")"}], "."]}]]}]}], ";"}]], "Input", CellLabel->"In[36]:=",ExpressionUUID->"1b768ef1-6736-408a-939d-8bdf459a8b84"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"result", "=", RowBox[{"SigmaReduce", "[", RowBox[{"mySum", ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"{", RowBox[{"m", "\[GreaterEqual]", "n"}], "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.949459159445014*^9, 3.9494591652118263`*^9}}, CellLabel->"In[37]:=",ExpressionUUID->"d47df31e-75cd-43c6-922d-d8226cc09476"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"1", "+", "m", "-", "n"}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"m"}, {"n"} }], ")"}], "."]}]], "+", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "-", "K", "-", "m", "+", "n"}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"K", "+", "m"}]}, {"n"} }], ")"}], "."]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", "m", "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["m", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], UnderscriptBox["m", "."]]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"m"}, {"n"} }], ")"}], "."]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "K", "+", "m", "-", "n"}], ")"}], " ", "x", " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"K", "+", "m"}]}, {"m"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["m", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], UnderscriptBox["m", "."]], " ", SuperscriptBox["x", UnderscriptBox["K", "."]]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"K", "+", "m"}]}, {"n"} }], ")"}], "."]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", "K", "-", "m", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["m", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], UnderscriptBox["m", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "K"], RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"m", "+", SubscriptBox["\[Iota]", "1"]}]}, {"m"} }], ")"}], "."], " ", SuperscriptBox["x", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}]}], ")"}]}], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"K", "+", "m"}]}, {"n"} }], ")"}], "."]}]], "+", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", "m", "+", "n"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["m", "."]], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], UnderscriptBox["m", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "K"], FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"m", "+", SubscriptBox["\[Iota]", "1"]}]}, {"m"} }], ")"}], "."], " ", SuperscriptBox["x", UnderscriptBox[ SubscriptBox["\[Iota]", "1"], "."]]}], SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"m", "+", SubscriptBox["\[Iota]", "1"]}]}, {"n"} }], ")"}], "."]]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "+", "n"}]]}]], "Output", CellChangeTimes->{3.9496412002682667`*^9}, CellLabel->"Out[37]=",ExpressionUUID->"564c9f55-1930-49c7-8c85-85bdbdebff35"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"mySum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ SuperscriptBox["k", "2"], " ", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], RowBox[{"X", "[", "i", "]"}]}]}]}]}], ";"}]], "Input", CellLabel->"In[38]:=",ExpressionUUID->"bbab1b34-c8d2-481d-b943-b26975ac4f79"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"res", "=", RowBox[{"SigmaReduce", "[", RowBox[{"mySum", ",", RowBox[{"XList", "\[Rule]", RowBox[{"{", "X", "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{3.949459235045752*^9}, CellLabel->"In[39]:=",ExpressionUUID->"f64f618c-c0b7-45a9-bd26-d5e3b0319275"], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "6"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], RowBox[{"X", "[", SubscriptBox["\[Iota]", "1"], "]"}]}], ")"}]}], "-", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], RowBox[{ SubscriptBox["\[Iota]", "1"], " ", RowBox[{"X", "[", SubscriptBox["\[Iota]", "1"], "]"}]}]}], ")"}]}], "+", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], RowBox[{ SubsuperscriptBox["\[Iota]", "1", "2"], " ", RowBox[{"X", "[", SubscriptBox["\[Iota]", "1"], "]"}]}]}], ")"}]}], "-", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], RowBox[{ SubsuperscriptBox["\[Iota]", "1", "3"], " ", RowBox[{"X", "[", SubscriptBox["\[Iota]", "1"], "]"}]}]}], ")"}]}]}]], "Output", CellChangeTimes->{3.949641201086989*^9}, CellLabel->"Out[39]=",ExpressionUUID->"f8ed4291-2584-445f-9878-52a2d352f1c2"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Refined creative telescoping", "Subsection", CellChangeTimes->{{3.9494179253969603`*^9, 3.9494179282777987`*^9}, { 3.949418254615407*^9, 3.9494182625278873`*^9}, {3.949464698740337*^9, 3.949464701579297*^9}},ExpressionUUID->"3dbb6229-2135-45b2-9c8a-\ 631400b86dd6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"S", "[", RowBox[{"2", ",", "k"}], "]"}], RowBox[{"SigmaBinomial", "[", RowBox[{"n", ",", "k"}], "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.949418411614193*^9, 3.9494184118178473`*^9}}, CellLabel->"In[40]:=",ExpressionUUID->"60a873db-1d04-4b53-a483-ebde6f8f1180"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."], " ", RowBox[{"S", "[", RowBox[{"2", ",", "k"}], "]"}]}]}]], "Output", CellChangeTimes->{3.949641201117765*^9}, CellLabel->"Out[40]=",ExpressionUUID->"2d5c48fc-30cc-40bc-ab5d-265c11a6013d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GenerateRecurrence", "[", "mySum", "]"}]], "Input", CellChangeTimes->{{3.949418425931036*^9, 3.94941842848846*^9}}, CellLabel->"In[41]:=",ExpressionUUID->"8e13fd13-3bf0-442c-a33f-808bba9dbe4b"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "-", RowBox[{"8", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], "2"], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"32", "+", RowBox[{"25", " ", "n"}], "+", RowBox[{"5", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"3", "+", "n"}], ")"}], "2"], " ", RowBox[{"SUM", "[", RowBox[{"3", "+", "n"}], "]"}]}]}], "\[Equal]", RowBox[{"-", "1"}]}], "}"}]], "Output", CellChangeTimes->{3.949641202699979*^9}, CellLabel->"Out[41]=",ExpressionUUID->"5338fd9f-71ee-4fc8-89c1-671672294e50"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GenerateRecurrence", "[", RowBox[{"mySum", ",", RowBox[{"SimplifyByExt", "->", "DepthNumber"}]}], "]"}]], "Input", CellChangeTimes->{{3.949418436051654*^9, 3.949418442474416*^9}}, CellLabel->"In[42]:=",ExpressionUUID->"ccd091d0-9168-48d2-b103-c05554a2ef65"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "\[Equal]", RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], RowBox[{"1", "+", "n", "-", SubscriptBox["\[Iota]", "1"]}]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], SubscriptBox["\[Iota]", "1"]]}]}], RowBox[{"1", "+", "n"}]]}]}], "}"}]], "Output", CellChangeTimes->{3.949641203831964*^9}, CellLabel->"Out[42]=",ExpressionUUID->"b06d2041-f51a-4c24-b997-f3d8941d8e1c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"S", "[", RowBox[{"2", ",", "1", ",", "k"}], "]"}], RowBox[{"SigmaBinomial", "[", RowBox[{"n", ",", "k"}], "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.949418269213472*^9, 3.949418287625703*^9}, { 3.949418347750684*^9, 3.9494183484848003`*^9}}, CellLabel->"In[43]:=",ExpressionUUID->"d85d751e-8653-4134-8432-5e019fed1b6b"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."], " ", RowBox[{"S", "[", RowBox[{"2", ",", "1", ",", "k"}], "]"}]}]}]], "Output", CellChangeTimes->{3.949641203870255*^9}, CellLabel->"Out[43]=",ExpressionUUID->"459eff6d-ef76-47b5-b3bc-c609b279541c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GenerateRecurrence", "[", RowBox[{"mySum", ",", RowBox[{"SimplifyByExt", "->", "None"}]}], "]"}]], "Input", CellChangeTimes->{{3.949418290811757*^9, 3.949418318097739*^9}, { 3.949418363145535*^9, 3.9494183684717083`*^9}}, CellLabel->"In[44]:=",ExpressionUUID->"9f03d605-85f2-4548-9c6e-b9be10d21b80"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "8"}], " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", "n"}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"4", " ", RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"12", "+", RowBox[{"5", " ", "n"}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{"3", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"80", "+", RowBox[{"53", " ", "n"}], "+", RowBox[{"9", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"302", "+", RowBox[{"256", " ", "n"}], "+", RowBox[{"73", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"7", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"3", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"4", "+", "n"}], ")"}], "3"], " ", RowBox[{"SUM", "[", RowBox[{"4", "+", "n"}], "]"}]}]}], "\[Equal]", RowBox[{"-", "1"}]}], "}"}]], "Output", CellChangeTimes->{3.949641207022908*^9}, CellLabel->"Out[44]=",ExpressionUUID->"7dbab2f5-2192-4b52-a656-6504f1b1f593"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GenerateRecurrence", "[", RowBox[{"mySum", ",", RowBox[{"SimplifyByExt", "->", "MinDepth"}]}], "]"}]], "Input", CellChangeTimes->{{3.9494183091122437`*^9, 3.949418326947838*^9}, { 3.94941837502703*^9, 3.9494183779186897`*^9}}, CellLabel->"In[45]:=",ExpressionUUID->"740ee600-3217-4fc8-925e-1c58d14a299d"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "\[Equal]", RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"-", "1"}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "0"}], "n"], FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], " ", RowBox[{"S", "[", RowBox[{"1", ",", SubscriptBox["\[Iota]", "1"]}], "]"}]}], RowBox[{"1", "+", "n", "-", SubscriptBox["\[Iota]", "1"]}]]}], ")"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, { SubscriptBox["\[Iota]", "1"]} }], ")"}], "."], " ", RowBox[{"S", "[", RowBox[{"1", ",", SubscriptBox["\[Iota]", "1"]}], "]"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "-", "n"}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}]}], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "3"]]}]}], "}"}]], "Output", CellChangeTimes->{3.949641208535116*^9}, CellLabel->"Out[45]=",ExpressionUUID->"ae82ae02-a5c0-4501-a2d9-a561a183c2ed"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The problem SigmaReduce", "Subsection", CellChangeTimes->{{3.827919006098606*^9, 3.827919009782981*^9}},ExpressionUUID->"1b3735ff-0a04-449a-8bdc-\ ddfd77c82269"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A1", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", "i"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], RowBox[{ RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"i", "^", "3"}]}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "k"}], "}"}]}], "]"}], "/", RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.8191765609624443`*^9, 3.819176601254879*^9}}, CellLabel->"In[46]:=",ExpressionUUID->"12a13c8e-f95a-4d87-831c-e1b38fcd95bd"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]], "Output", CellChangeTimes->{3.949641208588563*^9}, CellLabel->"Out[46]=",ExpressionUUID->"ff333d25-2365-4b15-ac8e-13ea2d949474"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"A2", "=", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], "-", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], RowBox[{"1", "+", "n"}]], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], RowBox[{"1", "+", "n"}]], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], "+", FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], RowBox[{"1", "+", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}]}]}], ";"}]], "Input", CellChangeTimes->{{3.819176449434576*^9, 3.819176467368972*^9}}, CellLabel->"In[47]:=",ExpressionUUID->"194ad338-f79d-4a70-b569-74b7828b87a2"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"A1", ",", "A2"}], "}"}], "/.", RowBox[{"n", "\[Rule]", "4"}]}]], "Input", CellChangeTimes->{{3.8279182816735363`*^9, 3.82791828982588*^9}}, CellLabel->"In[48]:=",ExpressionUUID->"f3825ade-d26c-4dfc-9f40-403c10bc236f"], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["14417", "6912"], ",", FractionBox["14417", "6912"]}], "}"}]], "Output", CellChangeTimes->{3.9496412086356792`*^9}, CellLabel->"Out[48]=",ExpressionUUID->"84ed0f3e-1271-4417-bad7-e88253c4b0ed"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", RowBox[{ RowBox[{"A1", "-", "A2"}], ",", "n"}], "]"}]], "Input", CellChangeTimes->{{3.8191764272075377`*^9, 3.819176433637816*^9}, { 3.8191764743297367`*^9, 3.819176492127537*^9}, {3.9493206787206993`*^9, 3.949320697283576*^9}}, CellLabel->"In[49]:=",ExpressionUUID->"e9680c2b-e365-493e-b1cd-34f3ba80a2c8"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.9496412110483627`*^9}, CellLabel->"Out[49]=",ExpressionUUID->"1a664b9a-64ed-4fe4-9808-b6b7919724b5"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"varL", "=", RowBox[{"Append", "[", RowBox[{ RowBox[{"Variables", "[", "A2", "]"}], ",", "A1"}], "]"}]}]], "Input", CellChangeTimes->{{3.819177999798284*^9, 3.81917805464017*^9}}, CellLabel->"In[50]:=",ExpressionUUID->"ce16e792-7e06-48f5-83ed-20474864e757"], Cell[BoxData[ RowBox[{"{", RowBox[{"n", ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}]}], "}"}]], "Output", CellChangeTimes->{3.9496412110935907`*^9}, CellLabel->"Out[50]=",ExpressionUUID->"0d2a40c0-82fc-4c40-8cb8-6f64c598bf04"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"varLRed", "=", RowBox[{"SigmaReduce", "[", RowBox[{"varL", ",", "n"}], "]"}]}]], "Input", CellChangeTimes->{{3.8191780569386044`*^9, 3.8191780652460938`*^9}, { 3.8191781369521112`*^9, 3.819178141149972*^9}, {3.8191783217720127`*^9, 3.819178322418404*^9}, {3.827918315557498*^9, 3.827918317280294*^9}}, CellLabel->"In[51]:=",ExpressionUUID->"e858dc7f-ad58-4869-a0bd-1af4267652f8"], Cell[BoxData[ RowBox[{"{", RowBox[{"n", ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], ",", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}], ",", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "-", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "-", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}], "}"}]], "Output", CellChangeTimes->{3.94964121366288*^9}, CellLabel->"Out[51]=",ExpressionUUID->"3dc59614-b4d2-46d6-b59e-9287f203c506"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"MapThread", "[", RowBox[{"Rule", ",", RowBox[{"{", RowBox[{"varL", ",", "varLRed"}], "}"}]}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.819178072888908*^9, 3.8191781194516487`*^9}, { 3.819178336757028*^9, 3.819178356396052*^9}, {3.827918319762723*^9, 3.827918321048726*^9}}, CellLabel->"In[52]:=",ExpressionUUID->"b0eaaa61-ba38-4fdc-a248-effadacadfee"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{"n", "\[Rule]", "n"}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "5"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "n"], FractionBox["1", SuperscriptBox["i", "4"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "4"]]}], "j"]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], SuperscriptBox["j", "2"]]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "\[Rule]", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "4"]]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "5"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", SuperscriptBox["i", "3"]]}], "j"]}], "k"]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "k"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "j"], FractionBox["1", "i"]}], SuperscriptBox["j", "3"]]}], "k"]}], "\[Rule]", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "-", RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}], ")"}]}]}]}]}, { RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", SuperscriptBox["i", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], "k"], FractionBox["1", "i"]}], ")"}]}], RowBox[{"1", "+", "k"}]]}], "\[Rule]", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "4"]]}], ")"}]}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], "-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubsuperscriptBox["\[Iota]", "1", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Iota]", "1"]]}], ")"}]}], RowBox[{ RowBox[{"-", "1"}], "-", "n"}]], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "4"]]}], SubscriptBox["\[Iota]", "1"]]}], "-", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], SubsuperscriptBox["\[Iota]", "1", "2"]]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "1"], "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubsuperscriptBox["\[Iota]", "2", "3"]]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Iota]", "2"], "=", "1"}], SubscriptBox["\[Iota]", "1"]], FractionBox["1", SubscriptBox["\[Iota]", "2"]]}], ")"}]}], SubscriptBox["\[Iota]", "1"]]}]}]}]} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.949641213726318*^9}, CellLabel-> "Out[52]//MatrixForm=",ExpressionUUID->"83800ec6-b314-4958-b6be-\ e255c72ecd38"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Part 3", "Title", CellChangeTimes->{{3.949324597535781*^9, 3.9493246008969584`*^9}},ExpressionUUID->"c748c669-5331-4ff0-97c3-\ f060bd8e100f"], Cell[BoxData[" "], "Input", CellChangeTimes->{ 3.949549843265448*^9},ExpressionUUID->"71705531-3e6e-4560-a9b3-\ 2b86826b5324"], Cell[CellGroupData[{ Cell["Unfair permutations", "Subtitle", CellChangeTimes->{{3.949549735957593*^9, 3.949549738916291*^9}, 3.949578133210594*^9},ExpressionUUID->"96461cb1-8314-40ab-8934-\ b7863f6cf054"], Cell[BoxData[ RowBox[{ RowBox[{"sumExpr", "=", RowBox[{ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ SuperscriptBox["j", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"i", "+", "j"}], ")"}], "2"]]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox["j", RowBox[{"i", "+", "j"}]]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ RowBox[{"j", " ", "k"}], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "j"}], ")"}], " ", RowBox[{"(", RowBox[{"i", "+", "k"}], ")"}]}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ RowBox[{"j", " ", "k"}], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "j"}], ")"}], " ", RowBox[{"(", RowBox[{"j", "+", "k"}], ")"}]}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ SuperscriptBox["k", "2"], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"j", "+", "k"}], ")"}]}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox["k", RowBox[{"i", "+", "j", "+", "k"}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ RowBox[{"j", " ", "k"}], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "j"}], ")"}], " ", RowBox[{"(", RowBox[{"i", "+", "j", "+", "k"}], ")"}]}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "3"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "2"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ RowBox[{"j", " ", "k"}], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "j"}], ")"}], " ", RowBox[{"(", RowBox[{"i", "+", "j", "+", "k"}], ")"}]}]]}]}]}], "+", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "k"}]], RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"i", "=", "1"}], RowBox[{ RowBox[{"-", "1"}], "+", "j"}]], FractionBox[ RowBox[{"j", " ", "k"}], RowBox[{ RowBox[{"(", RowBox[{"i", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"i", "+", "j", "+", "k"}], ")"}]}]]}]}]}]}]}], ";"}]], "Input", CellChangeTimes->{3.9495781594110727`*^9}, CellLabel->"In[5]:=",ExpressionUUID->"7898e03d-706d-4f6e-9007-cb987d39f624"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"res", "=", RowBox[{ RowBox[{"EvaluateMultiSum", "[", RowBox[{"sumExpr", ",", RowBox[{"{", "}"}], ",", RowBox[{"{", "n", "}"}], ",", RowBox[{"{", "3", "}"}]}], "]"}], "//", "Timing"}]}]], "Input", CellChangeTimes->{{3.949578164729678*^9, 3.9495781662642317`*^9}, { 3.949578570100666*^9, 3.9495785707426147`*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"d0e66e31-0b09-4033-9912-31c4f68ee547"], Cell[BoxData[ RowBox[{"{", RowBox[{"70.281776`", ",", RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", "1440"}], "+", RowBox[{"1391", " ", "n"}], "-", RowBox[{"162", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"72", " ", SuperscriptBox["n", "3"]}]}], "1080"], "+", RowBox[{ FractionBox["1", "6"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], "}"}], ",", "n"}], "]"}], "2"]}], "+", RowBox[{ FractionBox["1", "216"], " ", RowBox[{"(", RowBox[{"16", "+", RowBox[{"27", " ", "n"}], "-", RowBox[{"54", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "2"}], ",", "1"}], "}"}], "}"}], ",", "n"}], "]"}]}], "+", RowBox[{ FractionBox["1", "216"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "16"}], "+", RowBox[{"27", " ", "n"}], "-", RowBox[{"54", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], "}"}], ",", "n"}], "]"}]}], "-", RowBox[{ FractionBox["1", "3"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}], ",", "n"}], "]"}]}], "-", RowBox[{ FractionBox["1", "8"], " ", "n", " ", RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "1"}]}], "}"}], "}"}], ",", "n"}], "]"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], "+", RowBox[{ FractionBox["1", "8"], " ", "n", " ", RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"3", ",", RowBox[{"-", "1"}], ",", RowBox[{"-", "1"}]}], "}"}], "}"}], ",", "n"}], "]"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]]}], "+", RowBox[{ FractionBox["1", "12"], " ", "n", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"S", "[", RowBox[{ RowBox[{"-", "1"}], ",", "n"}], "]"}]}], "+", RowBox[{ FractionBox["1", "24"], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"22", " ", "n"}], "+", RowBox[{"2", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"16", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}], "-", RowBox[{ FractionBox["1", "24"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}], "2"]}], "+", RowBox[{ RowBox[{"CS", "[", RowBox[{ RowBox[{"{", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], "}"}], ",", "n"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "-", RowBox[{"15", " ", "n"}], "-", RowBox[{"4", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["n", "3"]}]}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"S", "[", RowBox[{"1", ",", "n"}], "]"}]}]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.949636534792941*^9}, CellLabel->"Out[6]=",ExpressionUUID->"fc7e7480-a438-4532-99cd-92c7fd784b76"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"GetFullDefinition", "[", RowBox[{"res", ",", "n"}], "]"}], "//", "ToSpecialFunction"}]], "Input", CellChangeTimes->{{3.949578538760497*^9, 3.9495785747260647`*^9}, { 3.949578628228121*^9, 3.949578630173524*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"4d514e12-1b7a-4fb5-b516-6e80800ff5fa"], Cell[BoxData[ RowBox[{"{", RowBox[{"70.281776`", ",", RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", "1440"}], "+", RowBox[{"1391", " ", "n"}], "-", RowBox[{"162", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"72", " ", SuperscriptBox["n", "3"]}]}], "1080"], "+", RowBox[{ FractionBox["1", "24"], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"22", " ", "n"}], "+", RowBox[{"2", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"16", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], ")"}]}], "-", RowBox[{ FractionBox["1", "24"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], ")"}], "2"]}], "+", RowBox[{ FractionBox["1", "12"], " ", "n", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Omicron]", "2"], "."]], SubscriptBox["\[Omicron]", "2"]]}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "5"}], "-", RowBox[{"15", " ", "n"}], "-", RowBox[{"4", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"16", " ", SuperscriptBox["n", "3"]}]}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", SubscriptBox["\[Omicron]", "2"]]}], ")"}]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "6"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}], "2"]}], "+", RowBox[{ FractionBox["1", "216"], " ", RowBox[{"(", RowBox[{"16", "+", RowBox[{"27", " ", "n"}], "-", RowBox[{"54", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}]}], "-", RowBox[{ FractionBox["1", "8"], " ", "n", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Omicron]", "2"], "."]], RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "216"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "16"}], "+", RowBox[{"27", " ", "n"}], "-", RowBox[{"54", " ", SuperscriptBox["n", "3"]}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"3", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}]}], "+", RowBox[{ FractionBox["1", "8"], " ", "n", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox["n", "."]], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], "n"], FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ SubscriptBox["\[Omicron]", "2"], "."]], RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"3", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], ")"}]}], "-", RowBox[{ FractionBox["1", "3"], " ", "n", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "3"], "=", "1"}], "n"], FractionBox[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{ SubscriptBox["\[Omicron]", "2"], "=", "1"}], SubscriptBox["\[Omicron]", "3"]], FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", RowBox[{"2", " ", SubscriptBox["\[Omicron]", "2"]}]}]]}], SubscriptBox["\[Omicron]", "3"]]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.9496366119867973`*^9}, CellLabel->"Out[7]=",ExpressionUUID->"f2ffc28c-d3f1-43a7-a19d-b248c0c0d631"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example: Finite Element Method", "Subtitle", CellChangeTimes->{{3.939615910725809*^9, 3.9396159131173973`*^9}},ExpressionUUID->"3fc7436c-b727-4b46-a22e-\ 2baa985f751e"], Cell[BoxData[ RowBox[{ RowBox[{"Jacobi", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"j", "=", "0"}], "k"], FractionBox[ RowBox[{ FractionBox[ RowBox[{ SubscriptBox[ RowBox[{"(", RowBox[{"-", "k"}], ")"}], "j"], " ", SubscriptBox[ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], "j"], " ", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"1", "-", "x"}], "2"], ")"}], UnderscriptBox["j", "."]]}], RowBox[{ RowBox[{"j", SubscriptBox["!", "."]}], " ", SubscriptBox[ RowBox[{"(", "2", ")"}], "j"]}]], " ", SubscriptBox[ RowBox[{"(", "2", ")"}], "k"]}], RowBox[{"k", SubscriptBox["!", "."]}]]}]}], ";"}]], "Input", CellChangeTimes->{{3.949579173235943*^9, 3.9495791820702353`*^9}}, CellLabel->"In[9]:=",ExpressionUUID->"a20f2869-00ca-4da0-a6d0-e0ecbee51037"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recP", "=", RowBox[{ RowBox[{ RowBox[{"GenerateRecurrence", "[", "Jacobi", "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "/.", RowBox[{"SUM", "\[Rule]", "P"}]}]}]], "Input", CellLabel->"In[10]:=",ExpressionUUID->"e25d9604-5186-4226-acfc-32122b241fd3"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"k", " ", RowBox[{"(", RowBox[{"2", "+", "k"}], ")"}], " ", RowBox[{"P", "[", "k", "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", " ", "k"}]}], ")"}], " ", "x", " ", RowBox[{"P", "[", RowBox[{"1", "+", "k"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "k"}], ")"}], " ", RowBox[{"P", "[", RowBox[{"2", "+", "k"}], "]"}]}]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.949636812070334*^9}, CellLabel->"Out[10]=",ExpressionUUID->"8e4bc562-bfce-4c34-b26a-cd88f909cbbc"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", "k"}], "+", "1"}], ")"}], "/", RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}]}], " ", RowBox[{"P", "[", "k", "]"}]}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{3.949579095051011*^9}, CellLabel->"In[11]:=",ExpressionUUID->"e4e443bd-38ab-41a4-b948-b701ee9ac781"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "1"}], "n"], FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", " ", "k"}]}], ")"}], " ", RowBox[{"P", "[", "k", "]"}]}], RowBox[{"1", "+", "k"}]]}]], "Output", CellChangeTimes->{3.949636863298943*^9}, CellLabel->"Out[11]=",ExpressionUUID->"825bb008-601b-4997-b5c1-aaac2a6f21ec"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", RowBox[{"mySum", ",", RowBox[{"{", RowBox[{"{", RowBox[{"recP", ",", RowBox[{"P", "[", "k", "]"}]}], "}"}], "}"}]}], "]"}]], "Input", CellLabel->"In[12]:=",ExpressionUUID->"52696a49-3875-407a-b653-79942e311a13"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "x"}]}], ")"}], " ", RowBox[{"P", "[", "1", "]"}]}], RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}]}]], "+", FractionBox[ RowBox[{"P", "[", "2", "]"}], RowBox[{"1", "-", "x"}]], "-", FractionBox[ RowBox[{"n", " ", RowBox[{"P", "[", "n", "]"}]}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}]}]], "+", FractionBox[ RowBox[{"P", "[", RowBox[{"1", "+", "n"}], "]"}], RowBox[{ RowBox[{"-", "1"}], "+", "x"}]]}]], "Output", CellChangeTimes->{3.9496368763626537`*^9}, CellLabel->"Out[12]=",ExpressionUUID->"5fe2bfb4-90ee-40b9-9255-1d61976a1f6c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["A=B", "Subtitle",ExpressionUUID->"767042f8-25d5-4c4f-a950-e29b507dabcd"], Cell[CellGroupData[{ Cell["lhs", "Section",ExpressionUUID->"3b88f14d-e0b5-4689-b8e0-b640d9d77ce6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recLHS", "=", RowBox[{"GenerateRecurrence", "[", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], SubscriptBox[ RowBox[{"(", SuperscriptBox[ RowBox[{"(", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."], ")"}], "4"], ")"}], SubscriptBox["k", "."]]}], "]"}]}]], "Input", CellLabel->"In[13]:=",ExpressionUUID->"0cd61c5e-fa1d-48b8-b3e2-33bf2e593a85"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"5", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"7", "+", RowBox[{"9", " ", "n"}], "+", RowBox[{"3", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], "3"], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}]}], "\[Equal]", "0"}], "}"}]], "Output",\ CellChangeTimes->{3.949637226601893*^9}, CellLabel->"Out[13]=",ExpressionUUID->"a302a6b6-af49-4f65-9bb0-12738756c73d"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["rhs", "Section", CellChangeTimes->{{3.939619479972589*^9, 3.93961948103078*^9}},ExpressionUUID->"d485dd37-564e-4b44-ba02-\ fddc888709d6"], Cell[BoxData[ RowBox[{ RowBox[{"innerSum", "=", RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"s", "=", "0"}], RowBox[{"1", "+", "n"}]], RowBox[{ SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ {"n"}, {"s"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"k", "+", "n"}]}, {"k"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{ RowBox[{"-", "k"}], "+", RowBox[{"2", " ", "n"}], "-", "s"}]}, {"n"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", GridBox[{ { RowBox[{"n", "+", "s"}]}, {"s"} }], ")"}], "."], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], UnderscriptBox[ RowBox[{"k", "+", "n", "+", "s"}], "."]]}]}]}], ";"}]], "Input", CellLabel->"In[14]:=",ExpressionUUID->"91ec4c1d-a257-4ff9-b180-0f7504ad3831"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"recK", "=", RowBox[{ RowBox[{ RowBox[{"GenerateRecurrence", "[", RowBox[{"innerSum", ",", "k"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "/.", RowBox[{"{", RowBox[{"SUM", "\[Rule]", "P"}], "}"}]}]}]], "Input", CellLabel->"In[15]:=",ExpressionUUID->"68cd3be9-3eff-4808-96ad-74cdfe125d3c"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"k", "-", "n"}], ")"}], "3"], " ", RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"P", "[", "k", "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{"2", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"k", "+", RowBox[{"2", " ", SuperscriptBox["k", "2"]}], "-", RowBox[{"3", " ", "n"}], "-", RowBox[{"6", " ", "k", " ", "n"}], "+", RowBox[{"3", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"P", "[", RowBox[{"1", "+", "k"}], "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "k"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{"k", "-", RowBox[{"3", " ", "n"}]}], ")"}], " ", RowBox[{"P", "[", RowBox[{"2", "+", "k"}], "]"}]}]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.949637267687273*^9}, CellLabel->"Out[15]=",ExpressionUUID->"e3b0b559-50a0-4292-ab98-a702542b3101"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"mySum", "=", RowBox[{"SigmaSum", "[", RowBox[{ RowBox[{"P", "[", "k", "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellLabel->"In[16]:=",ExpressionUUID->"c6d232c7-bda1-4cd8-96dd-246fcb189bac"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], RowBox[{"P", "[", "k", "]"}]}]], "Output", CellChangeTimes->{3.9496372747847652`*^9}, CellLabel->"Out[16]=",ExpressionUUID->"e54d2560-d8f5-493f-af32-bc162da81f76"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SigmaReduce", "[", RowBox[{"mySum", ",", RowBox[{"{", RowBox[{"recK", ",", RowBox[{"P", "[", "k", "]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.949579560578597*^9, 3.949579585365054*^9}}, CellLabel->"In[17]:=",ExpressionUUID->"97710d05-c154-4734-b611-8b7eff93177f"], Cell[BoxData[ RowBox[{ UnderoverscriptBox[ SuperscriptBox["\[Sum]", "."], RowBox[{"k", "=", "0"}], "n"], RowBox[{"P", "[", "k", "]"}]}]], "Output", CellChangeTimes->{3.9496372770890837`*^9}, CellLabel->"Out[17]=",ExpressionUUID->"5ad243a7-6c57-4a0c-948d-164e60ba2cd5"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"recK", StyleBox["N", FontColor->RGBColor[1, 0, 0]]}]], "=", RowBox[{ RowBox[{ RowBox[{"GenerateRecurrence", "[", RowBox[{"innerSum", ",", "k", ",", StyleBox[ RowBox[{"OneShiftIn", "->", "n"}], FontColor->RGBColor[1, 0, 0]]}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}], "/.", RowBox[{"SUM", "\[Rule]", "P"}]}]}]], "Input", CellLabel->"In[18]:=",ExpressionUUID->"aa8b9dac-22f3-4520-bdd6-576dd17274c1"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "k", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "5"}], " ", "k"}], "+", RowBox[{"12", " ", SuperscriptBox["k", "2"]}], "-", RowBox[{"10", " ", SuperscriptBox["k", "3"]}], "+", RowBox[{"3", " ", SuperscriptBox["k", "4"]}], "+", RowBox[{"3", " ", "n"}], "-", RowBox[{"32", " ", "k", " ", "n"}], "+", RowBox[{"42", " ", SuperscriptBox["k", "2"], " ", "n"}], "-", RowBox[{"16", " ", SuperscriptBox["k", "3"], " ", "n"}], "+", RowBox[{"15", " ", SuperscriptBox["n", "2"]}], "-", RowBox[{"57", " ", "k", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"33", " ", SuperscriptBox["k", "2"], " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"21", " ", SuperscriptBox["n", "3"]}], "-", RowBox[{"30", " ", "k", " ", SuperscriptBox["n", "3"]}], "+", RowBox[{"9", " ", SuperscriptBox["n", "4"]}]}], ")"}], " ", RowBox[{"P", "[", "k", "]"}]}], ")"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "k", "-", RowBox[{"3", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"6", "-", RowBox[{"8", " ", "k"}], "+", RowBox[{"3", " ", SuperscriptBox["k", "2"]}], "+", RowBox[{"12", " ", "n"}], "-", RowBox[{"8", " ", "k", " ", "n"}], "+", RowBox[{"6", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"P", "[", RowBox[{"1", "+", "k"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "k", "+", "n"}], ")"}], "3"], " ", RowBox[{"P", "[", RowBox[{ RowBox[{"1", "+", "n"}], ",", "k"}], "]"}]}]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.949637334884592*^9}, CellLabel->"Out[18]=",ExpressionUUID->"d6bb8de2-4730-478f-b429-da65f51a1c6c"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GenerateRecurrence", "[", RowBox[{"mySum", ",", "n", ",", RowBox[{"{", RowBox[{"{", RowBox[{"recK", ",", RowBox[{"P", "[", "k", "]"}]}], "}"}], "}"}], ",", "recKN"}], "]"}]], "Input", CellLabel->"In[19]:=",ExpressionUUID->"ff7c3fde-b40b-4f79-8400-99c120686a8e"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"5", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"7", "+", RowBox[{"9", " ", "n"}], "+", RowBox[{"3", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], "3"], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}]}], "\[Equal]", "0"}], "}"}]], "Output",\ CellChangeTimes->{3.949637378270782*^9}, CellLabel->"Out[19]=",ExpressionUUID->"ee862a35-fccc-4866-877f-e826385d382f"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData["recLHS"], "Input", CellLabel->"In[20]:=",ExpressionUUID->"e845f64a-01f9-4f32-bd98-f524381254ae"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "n"}], ")"}], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"5", "+", RowBox[{"4", " ", "n"}]}], ")"}], " ", RowBox[{"SUM", "[", "n", "]"}]}], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", " ", "n"}]}], ")"}], " ", RowBox[{"(", RowBox[{"7", "+", RowBox[{"9", " ", "n"}], "+", RowBox[{"3", " ", SuperscriptBox["n", "2"]}]}], ")"}], " ", RowBox[{"SUM", "[", RowBox[{"1", "+", "n"}], "]"}]}], "-", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "n"}], ")"}], "3"], " ", RowBox[{"SUM", "[", RowBox[{"2", "+", "n"}], "]"}]}]}], "\[Equal]", "0"}], "}"}]], "Output",\ CellChangeTimes->{3.949637382697596*^9}, CellLabel->"Out[20]=",ExpressionUUID->"db868dc3-8c34-4d62-9618-3bdd4ff6ef8c"] }, Open ]], Cell[BoxData[" "], "Input",ExpressionUUID->"2bb3b81d-39ab-461c-b103-797ff4450297"] }, Open ]] }, Open ]] }, Closed]] }, WindowSize->{1044, 626.25}, WindowMargins->{{-2.25, Automatic}, {Automatic, -22.5}}, Magnification:>1.5 Inherited, FrontEndVersion->"14.1 for Linux x86 (64-bit) (July 16, 2024)", StyleDefinitions->"Default.nb", ExpressionUUID->"d3171c60-c0cf-4345-8096-07ca9c785c1f" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "7899289383"->{ Cell[174456, 3864, 1963, 62, 571, "Print",ExpressionUUID->"7cd71215-6315-4c1f-a2b6-41d64fa13134", CellTags->"7899289383"]} } *) (*CellTagsIndex CellTagsIndex->{ {"7899289383", 340128, 9226} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 526, 10, 491, "Subsubsection",ExpressionUUID->"cd936624-eb53-4478-b2c8-5197555d265d"], Cell[1108, 34, 199, 3, 44, "Input",ExpressionUUID->"6184fcd3-f04e-49e5-8294-ce8a0e08da47"], Cell[1310, 39, 374, 6, 44, "Input",ExpressionUUID->"2c075768-366e-4a3e-be1f-e693a1710860"], Cell[CellGroupData[{ Cell[1709, 49, 552, 11, 105, "Input",ExpressionUUID->"09e10ab0-75ab-47fa-8de9-135f00ab5c40"], Cell[CellGroupData[{ Cell[2286, 64, 978, 26, 91, "Print",ExpressionUUID->"ae50fa83-f872-4430-9d83-5b6413a94f12"], Cell[3267, 92, 898, 24, 62, "Print",ExpressionUUID->"20512962-db41-4a5b-941f-e537bb0f5fe2"], Cell[4168, 118, 687, 18, 94, "Print",ExpressionUUID->"e9c0ec06-8bb5-49f2-84cf-8dab8d23ed54"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4916, 143, 151, 3, 144, "Title",ExpressionUUID->"e75bf771-6de1-4cf6-88bb-4df1e56cb17e"], Cell[5070, 148, 149, 2, 44, "Input",ExpressionUUID->"5a582f38-fe00-4067-90e5-9679c084050d"], Cell[CellGroupData[{ Cell[5244, 154, 170, 2, 114, "Section",ExpressionUUID->"92b73221-6725-4cc5-b951-f214acafd935"], Cell[CellGroupData[{ Cell[5439, 160, 161, 3, 79, "Subsection",ExpressionUUID->"dabbcba7-cdd2-460f-ab04-37fdbe1ebfcb"], Cell[5603, 165, 1634, 48, 224, "Input",ExpressionUUID->"49705659-a4e2-4d4d-974e-d352435df9e1"], Cell[CellGroupData[{ Cell[7262, 217, 330, 7, 44, "Input",ExpressionUUID->"43cd98a7-3d11-480e-ac6e-8fd966726e7d"], Cell[7595, 226, 1529, 48, 194, "Output",ExpressionUUID->"ea399b29-ee73-448f-ab4d-d42673858680"] }, Open ]], Cell[CellGroupData[{ Cell[9161, 279, 321, 5, 44, "Input",ExpressionUUID->"1c6215b7-9764-404a-a742-5945fa1f1c22"], Cell[9485, 286, 7187, 248, 584, "Output",ExpressionUUID->"4f8d8377-216e-4776-94dd-9f390d58cfe3"] }, Open ]], Cell[CellGroupData[{ Cell[16709, 539, 393, 8, 44, "Input",ExpressionUUID->"53d59f9a-ad82-48f7-b23b-d0644617fb96"], Cell[17105, 549, 202, 4, 34, "Print",ExpressionUUID->"104806a6-0ca9-4d41-a3aa-d868b92e38d7"], Cell[17310, 555, 718, 22, 75, "Output",ExpressionUUID->"fe340302-4735-485e-8690-fa0367f50df3"] }, Open ]], Cell[CellGroupData[{ Cell[18065, 582, 287, 5, 44, "Input",ExpressionUUID->"5c2c201f-3679-4950-8827-1ead90a4c342"], Cell[18355, 589, 851, 25, 132, "Output",ExpressionUUID->"ac47ca83-d2bd-49c0-8228-55b02f4c087c"] }, Open ]], Cell[CellGroupData[{ Cell[19243, 619, 367, 8, 44, "Input",ExpressionUUID->"e37c426f-a94a-4f48-8198-9a842ee9fd29"], Cell[19613, 629, 1477, 50, 148, "Output",ExpressionUUID->"80556b9e-b7a0-484c-bdd0-0d36f9934374"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[21139, 685, 91, 0, 79, "Subsection",ExpressionUUID->"75828a7d-5583-41ac-8a6b-88b7aeea3b74"], Cell[21233, 687, 1089, 32, 78, "Input",ExpressionUUID->"67b8a565-abde-4512-b681-30018ea1dd36"], Cell[CellGroupData[{ Cell[22347, 723, 411, 9, 44, "Input",ExpressionUUID->"2b6aa7cf-db86-40c1-a159-f75298f0ce5f"], Cell[22761, 734, 1721, 53, 117, "Output",ExpressionUUID->"439f749c-1f89-44e5-b247-20c28b3b8957"] }, Open ]], Cell[CellGroupData[{ Cell[24519, 792, 385, 8, 44, "Input",ExpressionUUID->"37427790-8ecf-47d0-96f0-49b8e144a929"], Cell[24907, 802, 675, 22, 76, "Output",ExpressionUUID->"e3376e21-0c22-436e-a6f4-fe395704f35a"] }, Open ]], Cell[CellGroupData[{ Cell[25619, 829, 301, 6, 44, "Input",ExpressionUUID->"68cd39e3-2a5d-4760-b9f4-6b12d57ae7d4"], Cell[25923, 837, 1026, 32, 104, "Output",ExpressionUUID->"f0ee9114-c891-4edd-a554-60030241dc51"] }, Open ]], Cell[CellGroupData[{ Cell[26986, 874, 374, 9, 44, "Input",ExpressionUUID->"025d336f-d5ea-4719-b236-fde3f43cea1d"], Cell[27363, 885, 672, 21, 104, "Output",ExpressionUUID->"073e7f4c-bf87-45cb-8e8b-c7fa597633e2"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[28096, 913, 222, 4, 99, "Section",ExpressionUUID->"e9d8a45e-77a3-4dbd-8ffe-ffe386a4f390"], Cell[28321, 919, 52085, 1015, 12993, "Input",ExpressionUUID->"36d1a0d5-e27b-46eb-bdc6-e5c576af20bb"], Cell[CellGroupData[{ Cell[80431, 1938, 660, 12, 74, "Input",ExpressionUUID->"5adadb75-9c29-49e4-8c9b-a5f2e82dcda2"], Cell[CellGroupData[{ Cell[81116, 1954, 337, 8, 34, "Print",ExpressionUUID->"d8854948-3c7d-4c05-978b-f5ba8aab79f6"], Cell[81456, 1964, 337, 8, 34, "Print",ExpressionUUID->"462e58a7-ce2d-4f14-8bc2-689ed6d05e7e"], Cell[81796, 1974, 337, 8, 34, "Print",ExpressionUUID->"a1b49246-6849-4ba7-a424-b0585fe8c997"], Cell[82136, 1984, 337, 8, 34, "Print",ExpressionUUID->"95f456c4-90a9-4f62-a51d-f69a1e09ad94"], Cell[82476, 1994, 337, 8, 34, "Print",ExpressionUUID->"5db666da-b813-493f-a1f3-6ba23a565104"], Cell[82816, 2004, 337, 8, 34, "Print",ExpressionUUID->"efbebf76-651f-41b1-bb1f-4fa131bc8d37"], Cell[83156, 2014, 339, 8, 34, "Print",ExpressionUUID->"e088f4c7-0697-4f27-9319-1242fbfdd5cb"], Cell[83498, 2024, 337, 8, 34, "Print",ExpressionUUID->"25785fe4-4c61-485a-885e-fb4ee2ddef5b"], Cell[83838, 2034, 209, 4, 34, "Print",ExpressionUUID->"514792c9-efdb-48b1-94dd-04332a09fa75"], Cell[84050, 2040, 209, 4, 34, "Print",ExpressionUUID->"bf00d469-4116-4d96-b2b6-f8cac17fcd36"], Cell[84262, 2046, 232, 5, 34, "Print",ExpressionUUID->"c2b5f774-c508-436f-9987-90b20453ca39"] }, Open ]], Cell[84509, 2054, 85819, 1687, 9054, "Output",ExpressionUUID->"c356f423-a1b0-4725-9408-4c1968e57f1e"] }, Open ]], Cell[CellGroupData[{ Cell[170365, 3746, 349, 7, 44, "Input",ExpressionUUID->"3509eb43-f0b8-4fb8-aeb4-d2ee362f7e06"], Cell[CellGroupData[{ Cell[170739, 3757, 339, 8, 34, "Print",ExpressionUUID->"289d926e-5db1-43b1-9ded-c4b2833444ba"], Cell[171081, 3767, 337, 8, 34, "Print",ExpressionUUID->"a900a013-db5a-44bd-a0f0-fa72e4db7c43"], Cell[171421, 3777, 337, 8, 34, "Print",ExpressionUUID->"ed010ce0-1456-4daa-86e3-00da3eac20e2"], Cell[171761, 3787, 336, 8, 34, "Print",ExpressionUUID->"bc51e80e-8163-401a-bb09-491409a3ca58"], Cell[172100, 3797, 337, 8, 34, "Print",ExpressionUUID->"e1116f1f-1338-4074-baa6-59c7ae25c2f9"], Cell[172440, 3807, 337, 8, 34, "Print",ExpressionUUID->"c86bc5fe-535f-4508-b525-a72ec7840934"], Cell[172780, 3817, 337, 8, 34, "Print",ExpressionUUID->"93154286-0e62-4d5c-b643-c0be9385b86f"], Cell[173120, 3827, 337, 8, 34, "Print",ExpressionUUID->"fdd77f12-207d-4aa6-81c9-56e6f8cf00b3"], Cell[173460, 3837, 209, 4, 34, "Print",ExpressionUUID->"be3fd421-9bc3-487d-9363-b2f0487a34f4"], Cell[173672, 3843, 533, 12, 34, "Print",ExpressionUUID->"225e76bf-bcfa-4e0d-99f0-08876fd6e5c9"], Cell[174208, 3857, 245, 5, 34, "Print",ExpressionUUID->"71e31919-22a0-4d14-a588-f86ffe5519d8"], Cell[174456, 3864, 1963, 62, 571, "Print",ExpressionUUID->"7cd71215-6315-4c1f-a2b6-41d64fa13134", CellTags->"7899289383"], Cell[176422, 3928, 209, 4, 34, "Print",ExpressionUUID->"f02c40e5-ba97-4c39-a476-c883b5518c0a"], Cell[176634, 3934, 232, 5, 34, "Print",ExpressionUUID->"b4dcf747-5d79-4ff3-9d40-9cff55508f82"] }, Open ]], Cell[176881, 3942, 17665, 527, 1245, "Output",ExpressionUUID->"19e45111-bcf3-4557-91f6-486b3f8eee9f"] }, Open ]], Cell[CellGroupData[{ Cell[194583, 4474, 894, 24, 162, "Input",ExpressionUUID->"5f8a7beb-04dc-43b6-a98d-60c3af2fb8fc"], Cell[195480, 4500, 12678, 418, 1045, "Output",ExpressionUUID->"940a00d4-1a77-423e-9223-117532f72c96"] }, Open ]], Cell[CellGroupData[{ Cell[208195, 4923, 417, 9, 44, "Input",ExpressionUUID->"3b5cebe6-6cd3-4870-9c84-b9641d5d439f"], Cell[208615, 4934, 5572, 188, 501, "Output",ExpressionUUID->"d6742d78-638b-4ada-8060-9374367ea7bc"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[214236, 5128, 224, 4, 99, "Section",ExpressionUUID->"88303882-1f2f-4f52-a441-9f31c1527641"], Cell[CellGroupData[{ Cell[214485, 5136, 169, 3, 79, "Subsection",ExpressionUUID->"a72b17d1-ab44-4073-a4ea-96381a4aafe4"], Cell[214657, 5141, 1723, 52, 224, "Input",ExpressionUUID->"6e2a3ddc-96ea-4346-a9f3-db9a18c3b252"], Cell[CellGroupData[{ Cell[216405, 5197, 444, 11, 78, "Input",ExpressionUUID->"c783cbdd-bb52-4984-985d-8ad7887415bb"], Cell[216852, 5210, 1886, 58, 70, "Output",ExpressionUUID->"68d73dc8-1d95-4e78-9da7-8eab59d7f9e1"] }, Open ]], Cell[CellGroupData[{ Cell[218775, 5273, 369, 8, 44, "Input",ExpressionUUID->"e3e91021-4933-402f-9b5f-e1c8ce4ca7f8"], Cell[219147, 5283, 546, 17, 70, "Output",ExpressionUUID->"0f76f322-bdd4-4570-810f-d128583f6e17"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[219742, 5306, 164, 3, 53, "Subsection",ExpressionUUID->"11c1b408-c5d8-46a7-9e2a-f89c68d1a94b"], Cell[219909, 5311, 967, 28, 89, "Input",ExpressionUUID->"fe7f6a19-52e8-4395-87df-8c97f04f3bff"], Cell[CellGroupData[{ Cell[220901, 5343, 369, 8, 46, "Input",ExpressionUUID->"28f72394-a9a2-446f-8da8-71bc889d64cf"], Cell[221273, 5353, 369, 11, 70, "Output",ExpressionUUID->"5b5e71e2-aed2-4bee-8a2b-089548a8d02e"] }, Open ]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[221715, 5372, 151, 3, 103, "Title",ExpressionUUID->"257c97cb-0133-4b0c-ad37-ed68ef03eb21"], Cell[CellGroupData[{ Cell[221891, 5379, 171, 3, 79, "Subsection",ExpressionUUID->"b8b85520-fee3-4236-9f2d-3be78b5a71dc"], Cell[222065, 5384, 1532, 43, 178, "Input",ExpressionUUID->"d5fa5f4d-5bd5-44b1-920f-548dd46257dc"], Cell[CellGroupData[{ Cell[223622, 5431, 213, 3, 44, "Input",ExpressionUUID->"af9459e8-488b-4b89-a56f-399d2ca97d1b"], Cell[223838, 5436, 3539, 117, 167, "Output",ExpressionUUID->"56ed1b9a-42d1-4a27-8230-b0a5b52cf3c3"] }, Open ]], Cell[CellGroupData[{ Cell[227414, 5558, 1085, 30, 107, "Input",ExpressionUUID->"c61d623f-db9c-4250-8c2b-a020c0c4e128"], Cell[228502, 5590, 956, 30, 128, "Output",ExpressionUUID->"51360d63-2fa8-4d92-beeb-e024c8e3ffbc"] }, Open ]], Cell[229473, 5623, 966, 27, 129, "Input",ExpressionUUID->"e5dee354-661a-42c7-9b14-febbb13829ce"], Cell[CellGroupData[{ Cell[230464, 5654, 266, 4, 44, "Input",ExpressionUUID->"960c8fb2-fe7d-4301-b7eb-88f6080d5dff"], Cell[230733, 5660, 2167, 69, 129, "Output",ExpressionUUID->"0de78a72-735e-4cb4-aa13-7e643d602bd4"] }, Open ]], Cell[232915, 5732, 552, 18, 80, "Input",ExpressionUUID->"f1a86c25-c098-441b-996a-1d303cf49943"], Cell[CellGroupData[{ Cell[233492, 5754, 216, 4, 44, "Input",ExpressionUUID->"cfc26608-d196-461b-9387-0fa737f711f3"], Cell[233711, 5760, 1588, 57, 89, "Output",ExpressionUUID->"6f007631-7747-4aec-84bf-fab37c1c6275"] }, Open ]], Cell[235314, 5820, 2457, 75, 214, "Input",ExpressionUUID->"d41dd618-a09c-4512-a5d5-4a674f3f4f13"], Cell[CellGroupData[{ Cell[237796, 5899, 239, 4, 44, "Input",ExpressionUUID->"cf18441e-9caf-4ef3-a334-4bc57e40444d"], Cell[238038, 5905, 5119, 175, 320, "Output",ExpressionUUID->"21da9b8c-beed-45c8-b353-ac6fd1144308"] }, Open ]], Cell[243172, 6083, 1292, 42, 134, "Input",ExpressionUUID->"1b768ef1-6736-408a-939d-8bdf459a8b84"], Cell[CellGroupData[{ Cell[244489, 6129, 375, 8, 44, "Input",ExpressionUUID->"d47df31e-75cd-43c6-922d-d8226cc09476"], Cell[244867, 6139, 4428, 166, 444, "Output",ExpressionUUID->"564c9f55-1930-49c7-8c85-85bdbdebff35"] }, Open ]], Cell[249310, 6308, 466, 14, 80, "Input",ExpressionUUID->"bbab1b34-c8d2-481d-b943-b26975ac4f79"], Cell[CellGroupData[{ Cell[249801, 6326, 299, 7, 44, "Input",ExpressionUUID->"f64f618c-c0b7-45a9-bd26-d5e3b0319275"], Cell[250103, 6335, 1669, 54, 88, "Output",ExpressionUUID->"f8ed4291-2584-445f-9878-52a2d352f1c2"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[251821, 6395, 277, 4, 79, "Subsection",ExpressionUUID->"3dbb6229-2135-45b2-9c8a-631400b86dd6"], Cell[CellGroupData[{ Cell[252123, 6403, 468, 12, 44, "Input",ExpressionUUID->"60a873db-1d04-4b53-a483-ebde6f8f1180"], Cell[252594, 6417, 422, 14, 83, "Output",ExpressionUUID->"2d5c48fc-30cc-40bc-ab5d-265c11a6013d"] }, Open ]], Cell[CellGroupData[{ Cell[253053, 6436, 219, 3, 44, "Input",ExpressionUUID->"8e13fd13-3bf0-442c-a33f-808bba9dbe4b"], Cell[253275, 6441, 1028, 32, 86, "Output",ExpressionUUID->"5338fd9f-71ee-4fc8-89c1-671672294e50"] }, Open ]], Cell[CellGroupData[{ Cell[254340, 6478, 290, 5, 44, "Input",ExpressionUUID->"ccd091d0-9168-48d2-b103-c05554a2ef65"], Cell[254633, 6485, 1324, 43, 109, "Output",ExpressionUUID->"b06d2041-f51a-4c24-b997-f3d8941d8e1c"] }, Open ]], Cell[CellGroupData[{ Cell[255994, 6533, 527, 13, 44, "Input",ExpressionUUID->"d85d751e-8653-4134-8432-5e019fed1b6b"], Cell[256524, 6548, 432, 14, 83, "Output",ExpressionUUID->"459eff6d-ef76-47b5-b3bc-c609b279541c"] }, Open ]], Cell[CellGroupData[{ Cell[256993, 6567, 334, 6, 44, "Input",ExpressionUUID->"9f03d605-85f2-4548-9c6e-b9be10d21b80"], Cell[257330, 6575, 1628, 51, 120, "Output",ExpressionUUID->"7dbab2f5-2192-4b52-a656-6504f1b1f593"] }, Open ]], Cell[CellGroupData[{ Cell[258995, 6631, 339, 6, 44, "Input",ExpressionUUID->"740ee600-3217-4fc8-925e-1c58d14a299d"], Cell[259337, 6639, 2147, 68, 163, "Output",ExpressionUUID->"ae82ae02-a5c0-4501-a2d9-a561a183c2ed"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[261533, 6713, 171, 3, 79, "Subsection",ExpressionUUID->"1b3735ff-0a04-449a-8bdc-ddfd77c82269"], Cell[CellGroupData[{ Cell[261729, 6720, 773, 22, 74, "Input",ExpressionUUID->"12a13c8e-f95a-4d87-831c-e1b38fcd95bd"], Cell[262505, 6744, 691, 22, 106, "Output",ExpressionUUID->"ff333d25-2365-4b15-ac8e-13ea2d949474"] }, Open ]], Cell[263211, 6769, 3404, 111, 265, "Input",ExpressionUUID->"194ad338-f79d-4a70-b569-74b7828b87a2"], Cell[CellGroupData[{ Cell[266640, 6884, 274, 6, 44, "Input",ExpressionUUID->"f3825ade-d26c-4dfc-9f40-403c10bc236f"], Cell[266917, 6892, 254, 6, 64, "Output",ExpressionUUID->"84ed0f3e-1271-4417-bad7-e88253c4b0ed"] }, Open ]], Cell[CellGroupData[{ Cell[267208, 6903, 362, 7, 44, "Input",ExpressionUUID->"e9680c2b-e365-493e-b1cd-34f3ba80a2c8"], Cell[267573, 6912, 151, 2, 50, "Output",ExpressionUUID->"1a664b9a-64ed-4fe4-9808-b6b7919724b5"] }, Open ]], Cell[CellGroupData[{ Cell[267761, 6919, 293, 6, 44, "Input",ExpressionUUID->"ce16e792-7e06-48f5-83ed-20474864e757"], Cell[268057, 6927, 3690, 124, 241, "Output",ExpressionUUID->"0d2a40c0-82fc-4c40-8cb8-6f64c598bf04"] }, Open ]], Cell[CellGroupData[{ Cell[271784, 7056, 417, 7, 44, "Input",ExpressionUUID->"e858dc7f-ad58-4869-a0bd-1af4267652f8"], Cell[272204, 7065, 12420, 389, 695, "Output",ExpressionUUID->"3dc59614-b4d2-46d6-b59e-9287f203c506"] }, Open ]], Cell[CellGroupData[{ Cell[284661, 7459, 430, 10, 44, "Input",ExpressionUUID->"b0eaaa61-ba38-4fdc-a248-effadacadfee"], Cell[285094, 7471, 19217, 546, 881, "Output",ExpressionUUID->"83800ec6-b314-4958-b6be-e255c72ecd38"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[304372, 8024, 151, 3, 103, "Title",ExpressionUUID->"c748c669-5331-4ff0-97c3-f060bd8e100f"], Cell[304526, 8029, 129, 3, 44, "Input",ExpressionUUID->"71705531-3e6e-4560-a9b3-2b86826b5324"], Cell[CellGroupData[{ Cell[304680, 8036, 188, 3, 78, "Subtitle",ExpressionUUID->"96461cb1-8314-40ab-8934-b7863f6cf054"], Cell[304871, 8041, 5657, 189, 325, "Input",ExpressionUUID->"7898e03d-706d-4f6e-9007-cb987d39f624"], Cell[CellGroupData[{ Cell[310553, 8234, 445, 10, 44, "Input",ExpressionUUID->"d0e66e31-0b09-4033-9912-31c4f68ee547"], Cell[311001, 8246, 4865, 159, 419, "Output",ExpressionUUID->"fc7e7480-a438-4532-99cd-92c7fd784b76"] }, Open ]], Cell[CellGroupData[{ Cell[315903, 8410, 331, 6, 44, "Input",ExpressionUUID->"4d514e12-1b7a-4fb5-b516-6e80800ff5fa"], Cell[316237, 8418, 7297, 230, 530, "Output",ExpressionUUID->"f2ffc28c-d3f1-43a7-a19d-b248c0c0d631"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[323583, 8654, 178, 3, 78, "Subtitle",ExpressionUUID->"3fc7436c-b727-4b46-a22e-2baa985f751e"], Cell[323764, 8659, 1007, 32, 111, "Input",ExpressionUUID->"a20f2869-00ca-4da0-a6d0-e0ecbee51037"], Cell[CellGroupData[{ Cell[324796, 8695, 293, 7, 44, "Input",ExpressionUUID->"e25d9604-5186-4226-acfc-32122b241fd3"], Cell[325092, 8704, 747, 23, 52, "Output",ExpressionUUID->"8e4bc562-bfce-4c34-b26a-cd88f909cbbc"] }, Open ]], Cell[CellGroupData[{ Cell[325876, 8732, 510, 15, 46, "Input",ExpressionUUID->"e4e443bd-38ab-41a4-b948-b701ee9ac781"], Cell[326389, 8749, 430, 13, 86, "Output",ExpressionUUID->"825bb008-601b-4997-b5c1-aaac2a6f21ec"] }, Open ]], Cell[CellGroupData[{ Cell[326856, 8767, 280, 7, 44, "Input",ExpressionUUID->"52696a49-3875-407a-b653-79942e311a13"], Cell[327139, 8776, 881, 31, 84, "Output",ExpressionUUID->"5fe2bfb4-90ee-40b9-9255-1d61976a1f6c"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[328069, 8813, 78, 0, 78, "Subtitle",ExpressionUUID->"767042f8-25d5-4c4f-a950-e29b507dabcd"], Cell[CellGroupData[{ Cell[328172, 8817, 77, 0, 99, "Section",ExpressionUUID->"3b88f14d-e0b5-4689-b8e0-b640d9d77ce6"], Cell[CellGroupData[{ Cell[328274, 8821, 538, 17, 77, "Input",ExpressionUUID->"0cd61c5e-fa1d-48b8-b3e2-33bf2e593a85"], Cell[328815, 8840, 1042, 33, 86, "Output",ExpressionUUID->"a302a6b6-af49-4f65-9bb0-12738756c73d"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[329906, 8879, 147, 3, 99, "Section",ExpressionUUID->"d485dd37-564e-4b44-ba02-fddc888709d6"], Cell[330056, 8884, 1171, 44, 78, "Input",ExpressionUUID->"91ec4c1d-a257-4ff9-b180-0f7504ad3831"], Cell[CellGroupData[{ Cell[331252, 8932, 346, 9, 44, "Input",ExpressionUUID->"68cd3be9-3eff-4808-96ad-74cdfe125d3c"], Cell[331601, 8943, 1319, 41, 117, "Output",ExpressionUUID->"e3b0b559-50a0-4292-ab98-a702542b3101"] }, Open ]], Cell[CellGroupData[{ Cell[332957, 8989, 282, 7, 44, "Input",ExpressionUUID->"c6d232c7-bda1-4cd8-96dd-246fcb189bac"], Cell[333242, 8998, 283, 7, 83, "Output",ExpressionUUID->"e54d2560-d8f5-493f-af32-bc162da81f76"] }, Open ]], Cell[CellGroupData[{ Cell[333562, 9010, 319, 7, 44, "Input",ExpressionUUID->"97710d05-c154-4734-b611-8b7eff93177f"], Cell[333884, 9019, 283, 7, 83, "Output",ExpressionUUID->"5ad243a7-6c57-4a0c-948d-164e60ba2cd5"] }, Open ]], Cell[CellGroupData[{ Cell[334204, 9031, 505, 15, 44, "Input",ExpressionUUID->"aa8b9dac-22f3-4520-bdd6-576dd17274c1"], Cell[334712, 9048, 2308, 69, 151, "Output",ExpressionUUID->"d6bb8de2-4730-478f-b429-da65f51a1c6c"] }, Open ]], Cell[CellGroupData[{ Cell[337057, 9122, 314, 8, 44, "Input",ExpressionUUID->"ff7c3fde-b40b-4f79-8400-99c120686a8e"], Cell[337374, 9132, 1042, 33, 86, "Output",ExpressionUUID->"ee862a35-fccc-4866-877f-e826385d382f"] }, Open ]], Cell[CellGroupData[{ Cell[338453, 9170, 111, 1, 44, "Input",ExpressionUUID->"e845f64a-01f9-4f32-bd98-f524381254ae"], Cell[338567, 9173, 1042, 33, 86, "Output",ExpressionUUID->"db868dc3-8c34-4d62-9618-3bdd4ff6ef8c"] }, Open ]], Cell[339624, 9209, 82, 0, 44, "Input",ExpressionUUID->"2bb3b81d-39ab-461c-b103-797ff4450297"] }, Open ]] }, Open ]] }, Closed]] } ] *)