Stirling number of the second kind: Specific values (formula 04.15.03.0056)

StirlingS2

$Mathematica Notation$

http://functions.wolfram.com/04.15.03.0056.01

Input Form

StirlingS2[m + 10, m] == (1/367873228800) (m (1 + m) (2 + m) (3 + m) (4 + m) (5 + m) (6 + m) (7 + m) (8 + m) (9 + m) (10 + m) (768 - 2288 m - 2068 m^2 + 11792 m^3 - 8195 m^4 - 8085 m^5 + 8778 m^6 + 6930 m^7 + 1485 m^8 + 99 m^9)) /; Element[m, Integers] && m >= 0

Standard Form

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MathML Form

<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["\[ScriptCapitalS]", StirlingS2] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mn> 10 </mn> </mrow> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 367873228800 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 99 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1485 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6930 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8778 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8085 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8195 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11792 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2068 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2288 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 768 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", 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'> 10 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 367873228800 </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 /> <ci> m </ci> <cn type='integer'> 9 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 10 </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'> 1485 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6930 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8778 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8085 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8195 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11792 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2068 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2288 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 768 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02