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http://functions.wolfram.com/04.15.03.0032.01
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StirlingS2[n, 7] == (1/5040) (7 - 21 2^n + 35 3^n - 35 4^n + 21 5^n - 7 6^n +
7^n - KroneckerDelta[n]) /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List["n", ",", "7"]], "]"]], "\[Equal]", " ", RowBox[List[FractionBox["1", "5040"], RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", SuperscriptBox["2", "n"]]], "+", RowBox[List["35", " ", SuperscriptBox["3", "n"]]], "-", RowBox[List["35", " ", SuperscriptBox["4", "n"]]], "+", RowBox[List["21", " ", SuperscriptBox["5", "n"]]], "-", RowBox[List["7", " ", SuperscriptBox["6", "n"]]], "+", SuperscriptBox["7", "n"], "-", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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["\[ScriptCapitalS]", StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 7 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 7 </mn> <mo> - </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <mn> 35 </mn> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 35 </mn> <mo> ⁢ </mo> <msup> <mn> 4 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <msup> <mn> 5 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <mn> 6 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <msup> <mn> 7 </mn> <mi> n </mi> </msup> </mrow> <mn> 5040 </mn> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> n </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> <ci> n </ci> <cn type='integer'> 7 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 6 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["n_", ",", "7"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["7", "-", RowBox[List["21", " ", SuperscriptBox["2", "n"]]], "+", RowBox[List["35", " ", SuperscriptBox["3", "n"]]], "-", RowBox[List["35", " ", SuperscriptBox["4", "n"]]], "+", RowBox[List["21", " ", SuperscriptBox["5", "n"]]], "-", RowBox[List["7", " ", SuperscriptBox["6", "n"]]], "+", SuperscriptBox["7", "n"], "-", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "5040"], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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