PartitionsP[n] == (1/(Pi Sqrt[2])) Sum[A[k, n] Sqrt[k] D[Sinh[(Pi Sqrt[2/3] Sqrt[n - 1/24])/k]/ (n - 1/24)^(1/2), n], {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}]

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<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> <mn> 1 </mn> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mn> 2 </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> <mrow> <mi> A </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> k </mi> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> n </mi> </mrow> </mfrac> </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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </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> <ci> A </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> n </ci> </bvar> <apply> <times /> <apply> <sinh /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <pi /> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 24 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 24 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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>

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2001-10-29