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/06.01.23.0044.01









  


  










Input Form





Sum[k!^2/((k + n)!^2 2^k), {k, 0, Infinity}] == Subscript[a, n] /; Element[n, Integers] && n > 0 && Subscript[a, 1] == 2 (Pi^2/12 - Log[2]^2/2) && Subscript[a, 2] == 2 (-3 + 2 Log[2] + 3 (Pi^2/12 - Log[2]^2/2)) && (Subscript[a, n] == (a[-2 + n] - 3 (1 + 2 (-2 + n)) (-2 + n) a[-1 + n] + (2 (-1 + 2 (-2 + n)) (-1 + n))/(-1 + n)!^2)/((2 - n) (-1 + n)^3) /; n > 2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], ")"]], "2"], " ", SuperscriptBox["2", "k"]]]]]], "\[Equal]", SubscriptBox["a", "n"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "12"], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"], "2"]]], ")"]]]]]], "\[And]", RowBox[List[SubscriptBox["a", "2"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "12"], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"], "2"]]], ")"]]]]]], ")"]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "n"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["a", "[", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], "]"]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]], " ", RowBox[List["a", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], ")"]], "2"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", "-", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "3"]]]]]], "/;", RowBox[List["n", ">", "2"]]]], ")"]]]]]]]]










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> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mn> 12 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mn> 12 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mfrac> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </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> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </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[SuperscriptBox[RowBox[List["(", RowBox[List["k_", "!"]], ")"]], "2"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["k_", "+", "n_"]], ")"]], "!"]], ")"]], "2"], " ", SuperscriptBox["2", "k_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SubscriptBox["a", "n"], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "12"], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"], "2"]]], ")"]]]]]], "&&", RowBox[List[SubscriptBox["a", "2"], "\[Equal]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "12"], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"], "2"]]], ")"]]]]]], ")"]]]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "n"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["a", "[", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], "]"]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]], " ", RowBox[List["a", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], ")"]], "2"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", "-", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "3"]]]]]], "/;", RowBox[List["n", ">", "2"]]]], ")"]]]]]]]]]]










Contributed by





Troy Kessler










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





2007-05-02





© 1998- Wolfram Research, Inc.