Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











StirlingS1






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/04.14.03.0032.01









  


  










Input Form





StirlingS2[n, 6] == (1/120) (-1)^n (-1 + n)! (HarmonicNumber[-1 + n]^5 - 10 HarmonicNumber[-1 + n]^3 HarmonicNumber[-1 + n, 2] + 20 HarmonicNumber[-1 + n]^2 HarmonicNumber[-1 + n, 3] - 20 HarmonicNumber[-1 + n, 2] HarmonicNumber[-1 + n, 3] + 15 HarmonicNumber[-1 + n] (HarmonicNumber[-1 + n, 2]^2 - 2 HarmonicNumber[-1 + n, 4]) + 24 HarmonicNumber[-1 + n, 5]) /; Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List["n", ",", "6"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "120"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "5"], "-", RowBox[List["10", " ", SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "3"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]]]], "+", RowBox[List["20", " ", SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "2"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "3"]], "]"]]]], "-", RowBox[List["20", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "3"]], "]"]]]], "+", RowBox[List["15", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "4"]], "]"]]]]]], ")"]]]], "+", RowBox[List["24", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "5"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "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> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 6 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 120 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 5 </mn> </msubsup> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 120 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["n_", ",", "6"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "120"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "5"], "-", RowBox[List["10", " ", SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "3"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]]]], "+", RowBox[List["20", " ", SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], "2"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "3"]], "]"]]]], "-", RowBox[List["20", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]], " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "3"]], "]"]]]], "+", RowBox[List["15", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "2"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "4"]], "]"]]]]]], ")"]]]], "+", RowBox[List["24", " ", RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "5"]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.