StirlingS2[m + 8, m] == (1/1393459200) (m (1 + m) (2 + m) (3 + m) (4 + m) (5 + m) (6 + m) (7 + m) (8 + m) (-144 + 404 m + 540 m^2 - 2345 m^3 + 840 m^4 + 3150 m^5 + 1260 m^6 + 135 m^7)) /; Element[m, Integers] && m >= 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> <mrow> <mi> m </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 1393459200 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 135 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1260 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3150 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 840 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2345 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 540 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 404 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 144 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </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> <apply> <plus /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 1393459200 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 135 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1260 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3150 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 840 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2345 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 540 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 404 </cn> <ci> m </ci> </apply> <cn type='integer'> -144 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02