StirlingS2[n, 8] == (1/40320) (KroneckerDelta[n] - 8 + 7 2^(2 + n) + 35 2^(1 + 2 n) - 56 3^n + 7 2^(2 + n) 3^n - 56 5^n - 8 7^n + 8^n) /; Element[n, Integers] && n >= 0

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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> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 8 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 40320 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mn> 5 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mn> 7 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <msup> <mn> 8 </mn> <mi> n </mi> </msup> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 35 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 40320 </cn> <apply> <plus /> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 8 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -8 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02