StirlingS1[m + 10, m] == (Pochhammer[m, 11]/367873228800) (10307425152 + 13175306672 m + 7220722828 m^2 + 2242194592 m^3 + 436886945 m^4 + 55598235 m^5 + 4634322 m^6 + 244530 m^7 + 7425 m^8 + 99 m^9) /; 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> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mn> 10 </mn> </mrow> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 367873228800 </mn> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 11 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], &quot;11&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 99 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7425 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 244530 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4634322 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 55598235 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 436886945 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2242194592 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7220722828 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13175306672 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 10307425152 </mn> </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> StirlingS1 </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 10 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 367873228800 </cn> <apply> <ci> Pochhammer </ci> <ci> m </ci> <cn type='integer'> 11 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 99 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7425 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 244530 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4634322 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 55598235 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 436886945 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2242194592 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7220722828 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 13175306672 </cn> <ci> m </ci> </apply> <cn type='integer'> 10307425152 </cn> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02