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StirlingS1




Mathematica Notation

Traditional Notation









Integer Functions > StirlingS1[n,m] > Specific values > Specialized values > For fixed m





http://functions.wolfram.com/04.14.03.0046.01









  


  








Input Form





StirlingS1[10, m] == -362880 KroneckerDelta[m - 1] + 1026576 KroneckerDelta[m - 2] - 1172700 KroneckerDelta[m - 3] + 723680 KroneckerDelta[m - 4] - 269325 KroneckerDelta[m - 5] + 63273 KroneckerDelta[m - 6] - 9450 KroneckerDelta[m - 7] + 870 KroneckerDelta[m - 8] - 45 KroneckerDelta[m - 9] + KroneckerDelta[m - 10] /; Element[m, Integers] && m >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["10", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "362880"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]]]], "+", RowBox[List["1026576", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "-", RowBox[List["1172700", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "3"]], "]"]]]], "+", RowBox[List["723680", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "4"]], "]"]]]], "-", RowBox[List["269325", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "5"]], "]"]]]], "+", RowBox[List["63273", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "6"]], "]"]]]], "-", RowBox[List["9450", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "7"]], "]"]]]], "+", RowBox[List["870", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "8"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "9"]], "]"]]]], "+", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "10"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]








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[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mn> 10 </mn> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </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> 10 </mn> </mrow> </msub> <mo> - </mo> <mrow> <mn> 45 </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> 9 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mn> 870 </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> 8 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 9450 </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> </mrow> <mo> + </mo> <mrow> <mn> 63273 </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> 6 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 269325 </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> </mrow> <mo> + </mo> <mrow> <mn> 723680 </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> 4 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 1172700 </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> 3 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mn> 1026576 </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> 2 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 362880 </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> 1 </mn> </mrow> </msub> </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> <cn type='integer'> 10 </cn> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -9 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 870 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9450 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -7 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 63273 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 269325 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -5 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 723680 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1172700 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1026576 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 362880 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </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["StirlingS1", "[", RowBox[List["10", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "362880"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]]]], "+", RowBox[List["1026576", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "-", RowBox[List["1172700", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "3"]], "]"]]]], "+", RowBox[List["723680", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "4"]], "]"]]]], "-", RowBox[List["269325", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "5"]], "]"]]]], "+", RowBox[List["63273", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "6"]], "]"]]]], "-", RowBox[List["9450", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "7"]], "]"]]]], "+", RowBox[List["870", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "8"]], "]"]]]], "-", RowBox[List["45", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "9"]], "]"]]]], "+", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "10"]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02