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Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Identities > Recurrence identities > Generalized Cauchy summation





http://functions.wolfram.com/06.04.17.0004.01









  


  










Input Form





BoxData[\(\(\(Sum[\(\(\(\((\(-1\))\)^\(Sum[\(\(Subscript[\(j, \(2 * h\)\)]\), \({h, 1, p}\)\)]\)\) * \(Multinomial[\(\(m - \(Sum[\(\(Subscript[\(j, n\)]\), \({h, 1, n}\)\)]\)\), \(Subscript[\(j, 1\)]\), \(Subscript[\(j, 2\)]\), …, \(Subscript[\(j, n\)]\)\)]\)\), \({\(Subscript[\(j, n\)]\), 0, m}\), \({\(Subscript[\(j, 1\)]\), 0, m}\), …, \({\(Subscript[\(j, 2\)]\), 0, m}\)\)]\)  \(1 - \(KroneckerDelta[\(Mod[\(n, 2\)]\)]\)\)\)/;\(\(m ∈ Integers\) && \(m ≥ 0\) && \(n ∈ Integers\) && \(n ≥ 1\) && \(p ∈ Integers\) && \(p ≥ 1\)\)\)]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], "m"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "p"], SubscriptBox["j", RowBox[List["2", "h"]]]]]], RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "n"], SubscriptBox["j", "n"]]]]], ",", SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["j", "n"]]], "]"]]]]]]]]]]]], "\[Equal]", " ", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "1"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> j </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> h </mi> </mrow> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> j </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#948; </mi> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> j </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> h </mi> </mrow> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> j </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#948; </mi> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j_", "1"], "=", "0"]], "m_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j_", "2"], "=", "0"]], "m_"], RowBox[List["\[Ellipsis]_", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j_", "n_"], "=", "0"]], "m_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h_", "=", "1"]], "p_"], SubscriptBox["j_", RowBox[List["2", " ", "h_"]]]]]], " ", RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["m_", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h_", "=", "1"]], "n_"], SubscriptBox["j_", "n_"]]]]], ",", SubscriptBox["j_", "1"], ",", SubscriptBox["j_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["j_", "n_"]]], "]"]]]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]]]]]]]]]]










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





2002-12-18





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