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

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/06.04.17.0002.01









  


  










Input Form





Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] == Sum[\[Ellipsis] Sum[KroneckerDelta[n - Sum[Subscript[k, j], {j, 1, m}], 0] Multinomial[Subscript[k, 1], Subscript[k, 2], \[Ellipsis], Subscript[k, m]] Multinomial[Subscript[n, 1] - Subscript[k, 1], Subscript[n, 2] - Subscript[k, 2], \[Ellipsis], Subscript[n, m] - Subscript[k, m]], {Subscript[k, m], 0, n}], {Subscript[k, 1], 0, n}, {Subscript[k, 2], 0, n}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "m"], "=", "0"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], SubscriptBox["k", "j"]]]]], ",", "0"]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["k", "m"]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["k", "1"]]], ",", RowBox[List[SubscriptBox["n", "2"], "-", SubscriptBox["k", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["n", "m"], "-", SubscriptBox["k", "m"]]]]], "]"]]]]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> m </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> k </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;k&quot;, &quot;m&quot;]]], &quot;;&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;k&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> <mo> - </mo> <msub> <mi> k </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> n </mi> <mi> m </mi> </msub> <mo> - </mo> <msub> <mi> k </mi> <mi> m </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;m&quot;]]], &quot;;&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;]]]]], &quot;,&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;2&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;m&quot;], &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;m&quot;]]]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "m"], "=", "0"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], SubscriptBox["k", "j"]]]]], ",", "0"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["k", "m"]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[RowBox[List[SubscriptBox["nn", "1"], "-", SubscriptBox["k", "1"]]], ",", RowBox[List[SubscriptBox["nn", "2"], "-", SubscriptBox["k", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["nn", "m"], "-", SubscriptBox["k", "m"]]]]], "]"]]]]]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.