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StirlingS2




Mathematica Notation

Traditional Notation









Integer Functions > StirlingS2[n,m] > Specific values > Specialized values > For fixed m





http://functions.wolfram.com/04.15.03.0045.01









  


  








Input Form





StirlingS2[9, m] == ((-1)^(m - 1)/(m - 1)!) (KroneckerDelta[m - 1] + 255 (1 - m) KroneckerDelta[m - 2] + 3025 (-2 + m) (-1 + m) KroneckerDelta[-3 + m] - 7770 (-3 + m) (-2 + m) (-1 + m) KroneckerDelta[-4 + m] + 6951 Pochhammer[m - 4, 4] KroneckerDelta[-5 + m] - 2646 Pochhammer[m - 5, 5] KroneckerDelta[-6 + m] + 462 Pochhammer[m - 6, 6] KroneckerDelta[-7 + m] - 36 Pochhammer[m - 7, 7] KroneckerDelta[-8 + m] + Pochhammer[m - 8, 8] KroneckerDelta[-9 + m]) /; Element[m, Integers] && m >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List["9", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]], "+", RowBox[List["255", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "+", RowBox[List["3025", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], "]"]]]], "-", RowBox[List["7770", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], "]"]]]], " ", "+", RowBox[List["6951", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "4"]], ",", "4"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "5"]], "+", "m"]], "]"]]]], "-", RowBox[List["2646", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "5"]], ",", "5"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "6"]], "+", "m"]], "]"]]]], "+", RowBox[List["462", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "6"]], ",", "6"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "7"]], "+", "m"]], "]"]]]], "-", RowBox[List["36", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "7"]], ",", "7"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "8"]], "+", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "8"]], ",", "8"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "9"]], "+", "m"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







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</mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> - </mo> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 7 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;7&quot;]], &quot;)&quot;]], &quot;7&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 8 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2646 </mn> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;5&quot;]], &quot;)&quot;]], &quot;5&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; 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</mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 3 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 9 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;8&quot;]], &quot;)&quot;]], &quot;8&quot;], Pochhammer] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 462 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 7 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;6&quot;]], &quot;)&quot;]], &quot;6&quot;], Pochhammer] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 6951 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 5 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;4&quot;]], &quot;)&quot;]], &quot;4&quot;], Pochhammer] </annotation> </semantics> </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> <cn type='integer'> 9 </cn> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 255 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -8 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2646 </cn> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -6 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7770 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3025 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -9 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -8 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 462 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -7 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -6 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6951 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -5 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -4 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["9", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]], "+", RowBox[List["255", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "+", RowBox[List["3025", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], "]"]]]], "-", RowBox[List["7770", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], "]"]]]], "+", RowBox[List["6951", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "4"]], ",", "4"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "5"]], "+", "m"]], "]"]]]], "-", RowBox[List["2646", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "5"]], ",", "5"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "6"]], "+", "m"]], "]"]]]], "+", RowBox[List["462", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "6"]], ",", "6"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "7"]], "+", "m"]], "]"]]]], "-", RowBox[List["36", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "7"]], ",", "7"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "8"]], "+", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "8"]], ",", "8"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "9"]], "+", "m"]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02