|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/04.15.03.0030.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
StirlingS2[n, 5] == (1/120) (5 - 5 2^(1 + n) + 10 3^n - 5 4^n + 5^n -
KroneckerDelta[n]) /; Element[n, Integers] && n >= 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List["n", ",", "5"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "120"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["5", " ", SuperscriptBox["2", RowBox[List["1", "+", "n"]]]]], "+", RowBox[List["10", " ", SuperscriptBox["3", "n"]]], "-", RowBox[List["5", " ", SuperscriptBox["4", "n"]]], "+", SuperscriptBox["5", "n"], "-", " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> 5 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 5 </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> 10 </mn> <mo> ⁢ </mo> <msup> <mn> 3 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mn> 4 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <msup> <mn> 5 </mn> <mi> n </mi> </msup> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mn> 120 </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'> 5 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 10 </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'> 5 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 120 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["n_", ",", "5"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "120"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["5", " ", SuperscriptBox["2", RowBox[List["1", "+", "n"]]]]], "+", RowBox[List["10", " ", SuperscriptBox["3", "n"]]], "-", RowBox[List["5", " ", SuperscriptBox["4", "n"]]], "+", SuperscriptBox["5", "n"], "-", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|