Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Factorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Summation > Infinite summation > Parameter-containing sums





http://functions.wolfram.com/06.01.23.0057.01









  


  










Input Form





Sum[(4 k)!/((4 k + 3 n + 1)! 9^k), {k, 0, Infinity}] == ((1 - Sqrt[3])^(3 n) (Sqrt[3] Log[Sqrt[3]/(Sqrt[3] - 1)]))/(4 (3 n)!) + (1/4) Sum[(Sqrt[3] (Sqrt[3] + 1)^j - (1 - Sqrt[3])^j Sqrt[3])/ ((-3 n + j) (3 n)!), {j, 0, 3 n - 1}] + ((Sqrt[3] + 1)^(3 n) (Sqrt[3] Log[(Sqrt[3] + 1)/Sqrt[3]]))/(4 (3 n)!) + (1/2) (Sum[-((3 (-8)^j (4 - 9 n + 9 j))/((1 - 3 n + 3 j) (2 - 3 n + 3 j) (3 n)!)), {j, 0, n - 1}] + ((-8)^n Pi)/((3 n (3 n - 1)!) (2 Sqrt[3]))) /; Element[n, Integers] && n >= 1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]], " ", SuperscriptBox["9", "k"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], RowBox[List["3", " ", "n"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[SqrtBox["3"], RowBox[List[SqrtBox["3"], "-", "1"]]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["3", " ", "n"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List[SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], "j"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], "j"], " ", SqrtBox["3"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "n"]], "+", "j"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], RowBox[List["3", " ", "n"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[SqrtBox["3"], "+", "1"]], SqrtBox["3"]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["-", FractionBox[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["9", " ", "n"]], "+", RowBox[List["9", " ", "j"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "n"]], "+", RowBox[List["3", " ", "j"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "n"]], "+", RowBox[List["3", " ", "j"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "n"], " ", "\[Pi]"]], RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", SqrtBox["3"]]], ")"]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mn> 9 </mn> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <msqrt> <mn> 3 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msqrt> <mn> 3 </mn> </msqrt> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 9 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> -8 </cn> <ci> j </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -8 </cn> <ci> n </ci> </apply> <pi /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k_"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k_"]], "+", RowBox[List["3", " ", "n_"]], "+", "1"]], ")"]], "!"]], " ", SuperscriptBox["9", "k_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], RowBox[List["3", " ", "n"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[SqrtBox["3"], RowBox[List[SqrtBox["3"], "-", "1"]]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["3", " ", "n"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List[SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], "j"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], "j"], " ", SqrtBox["3"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "n"]], "+", "j"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], RowBox[List["3", " ", "n"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[SqrtBox["3"], "+", "1"]], SqrtBox["3"]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["-", FractionBox[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["9", " ", "n"]], "+", RowBox[List["9", " ", "j"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "n"]], "+", RowBox[List["3", " ", "j"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "n"]], "+", RowBox[List["3", " ", "j"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "n"], " ", "\[Pi]"]], RowBox[List[RowBox[List["(", RowBox[List["3", " ", "n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", SqrtBox["3"]]], ")"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]]]










Contributed by





Troy Kessler










Date Added to functions.wolfram.com (modification date)





2007-05-02





© 1998- Wolfram Research, Inc.