Sum[k!^3/(k + 2 n + 1)!^3, {k, 0, Infinity}] == ((3 n)!/(n!^3 (2 n)!^3)) Sum[((-1)^j (5 - 32 n + 56 n^2 + 32 j - 112 n j + 56 j^2) (-1 + n - j)!^3)/(4 (-1 + 2 n - 2 j)^2 (3 n - 3 j)!), {j, 0, n - 1}] + (((-1)^n (3 n)!) Zeta[3])/(n!^3 (2 n)!^3) /; Element[n, Integers] && n > 0

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</mo> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mi> j </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 112 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <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> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ! 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Troy Kessler

2007-05-02