Stirling number of the second kind: Specific values (formula 04.15.03.0055)

StirlingS2

$Mathematica Notation$

http://functions.wolfram.com/04.15.03.0055.01

Input Form

StirlingS2[m + 9, m] == (1/2786918400) (m^2 (1 + m)^2 (2 + m) (3 + m) (4 + m) (5 + m) (6 + m) (7 + m) (8 + m) (9 + m) (-144 + 548 m - 648 m^2 - 17 m^3 + 465 m^4 + 165 m^5 + 15 m^6)) /; 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> 9 </mn> </mrow> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2786918400 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <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> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 165 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 465 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 17 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 648 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 548 </mn> <mo> ⁢ </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> ∈ </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'> 9 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2786918400 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </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 /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 165 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 465 </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'> 17 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 648 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 548 </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>

Rule Form

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

2007-05-02