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
 











PartitionsP






Mathematica Notation

Traditional Notation









Integer Functions > PartitionsP[n] > Series representations > Generalized power series





http://functions.wolfram.com/04.16.06.0003.01









  


  










Input Form





PartitionsP[n] == (Pi^2/(9 Sqrt[3])) Sum[(A[k, n]/k^(5/2)) Hypergeometric0F1[5/2, ((-(1/24) + n) Pi^2)/ (6 k^2)], {k, 1, Infinity}] /; A[k, n] == Sum[KroneckerDelta[GCD[h, k], 1] Exp[Pi I Sum[(j/k) ((h j)/k - Floor[(h j)/k] - 1/2), {j, 1, k - 1}] - (2 Pi I h n)/k], {h, 1, k}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PartitionsP", "[", "n", "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["9", " ", SqrtBox["3"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["A", "[", RowBox[List["k", ",", "n"]], "]"]], SuperscriptBox["k", RowBox[List["5", "/", "2"]]]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "24"]]], "+", "n"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["6", " ", SuperscriptBox["k", "2"]]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["A", "[", RowBox[List["k", ",", "n"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "k"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["h", ",", "k"]], "]"]], ",", "1"]], "]"]], RowBox[List["Exp", "[", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List["j", " "]], "k"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["h", " ", "j"]], "k"], "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["h", " ", "j"]], "k"], "]"]], "-", FractionBox["1", "2"]]], ")"]]]]]]]], "-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "h", " ", "n"]], "k"]]], "]"]]]]]]]]]]]]










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> <semantics> <mi> p </mi> <annotation encoding='Mathematica'> TagBox[&quot;p&quot;, PartitionsP] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mi> k </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </mo> <mo> ; </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;0&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;5&quot;, &quot;2&quot;], Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, FractionBox[&quot;1&quot;, &quot;24&quot;]]], &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[Pi]&quot;, &quot;2&quot;]]], RowBox[List[&quot;6&quot;, &quot; &quot;, SuperscriptBox[&quot;k&quot;, &quot;2&quot;]]]], Hypergeometric0F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> h </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mi> j </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> h </mi> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mi> k </mi> </mfrac> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> h </mi> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mi> k </mi> </mfrac> <mo> &#8971; </mo> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> h </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PartitionsP </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> A </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric0F1 </ci> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 24 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> A </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> h </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <gcd /> <ci> h </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <exp /> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> <apply> <plus /> <apply> <times /> <ci> h </ci> <ci> j </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> h </ci> <ci> j </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> h </ci> <ci> n </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PartitionsP", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["A", "[", RowBox[List["k", ",", "n"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "24"]]], "+", "n"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["6", " ", SuperscriptBox["k", "2"]]]]]], "]"]]]], SuperscriptBox["k", RowBox[List["5", "/", "2"]]]]]]]], RowBox[List["9", " ", SqrtBox["3"]]]], "/;", RowBox[List[RowBox[List["A", "[", RowBox[List["k", ",", "n"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "k"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["h", ",", "k"]], "]"]], ",", "1"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List["j", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["h", " ", "j"]], "k"], "-", RowBox[List["Floor", "[", FractionBox[RowBox[List["h", " ", "j"]], "k"], "]"]], "-", FractionBox["1", "2"]]], ")"]]]], "k"]]]]], "-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "h", " ", "n"]], "k"]]]]]]]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.