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Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/06.04.17.0008.01









  


  










Input Form





Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] == (Pochhammer[Subscript[n, l] + 1, p]/ Pochhammer[Sum[Subscript[n, j], {j, 1, m}] + 1, p]) Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, l - 1], Subscript[n, l] + p, Subscript[n, l + 1], \[Ellipsis], Subscript[n, m]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["n", "l"], "+", "1"]], ",", "p"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], SubscriptBox["n", "j"]]], "+", "1"]], ",", "p"]], "]"]]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", RowBox[List["l", "-", "1"]]], ",", RowBox[List[SubscriptBox["n", "l"], "+", "p"]], ",", SubscriptBox["n", RowBox[List["l", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </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;n&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;;&quot;, SubscriptBox[&quot;n&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> l </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;l&quot;], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;p&quot;], Pochhammer] </annotation> </semantics> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[UnderoverscriptBox[&quot;\[Sum]&quot;, RowBox[List[&quot;j&quot;, &quot;=&quot;, &quot;1&quot;]], &quot;m&quot;], SubscriptBox[&quot;n&quot;, &quot;j&quot;]]], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;p&quot;], Pochhammer] </annotation> </semantics> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> p </mi> </mrow> <mo> ; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mrow> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mrow> <msub> <mi> n </mi> <mi> l </mi> </msub> <mo> + </mo> <mi> p </mi> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mrow> <mi> l </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </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;n&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;], &quot;+&quot;, &quot;p&quot;]], &quot;;&quot;, SubscriptBox[&quot;n&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;l&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;,&quot;, RowBox[List[SubscriptBox[&quot;n&quot;, &quot;l&quot;], &quot;+&quot;, &quot;p&quot;]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, RowBox[List[&quot;l&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> p </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <plus /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> <ci> p </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <plus /> <ci> l </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </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]", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["nn", "l"], "+", "1"]], ",", "p"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["nn", "1"], ",", SubscriptBox["nn", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", RowBox[List["l", "-", "1"]]], ",", RowBox[List[SubscriptBox["nn", "l"], "+", "p"]], ",", SubscriptBox["n", RowBox[List["l", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["nn", "m"]]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], SubscriptBox["nn", "j"]]], "+", "1"]], ",", "p"]], "]"]]]]]]]










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





2007-05-02





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