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http://functions.wolfram.com/06.04.06.0003.01
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Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis],
Subscript[n, m]] == SeriesTerm[Sum[Subscript[t, k], {k, 1, m}]^n,
{Subscript[t, 1], 0, Subscript[n, 1]}, {Subscript[t, 2], 0,
Subscript[n, 2]}, \[Ellipsis], {Subscript[t, m], 0, Subscript[n, m]}] /;
n == Sum[Subscript[n, k], {k, 1, m}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Equal]", RowBox[List["SeriesTerm", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["t", "k"]]], ")"]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "1"], ",", "0", ",", SubscriptBox["n", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "2"], ",", "0", ",", SubscriptBox["n", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "m"], ",", "0", ",", SubscriptBox["n", "m"]]], "}"]]]], "]"]]]], "/;", RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["n", "k"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <msubsup> <mi> t </mi> <mn> 1 </mn> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </msubsup> <mo> , </mo> <msubsup> <mi> t </mi> <mn> 2 </mn> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mi> t </mi> <mi> m </mi> <msub> <mi> n </mi> <mi> m </mi> </msub> </msubsup> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> t </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <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> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <msubsup> <mi> t </mi> <mn> 1 </mn> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </msubsup> <mo> , </mo> <msubsup> <mi> t </mi> <mn> 2 </mn> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mi> t </mi> <mi> m </mi> <msub> <mi> n </mi> <mi> m </mi> </msub> </msubsup> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> t </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| 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[RowBox[List["SeriesTerm", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["t", "k"]]], ")"]], "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "1"], ",", "0", ",", SubscriptBox["nn", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "2"], ",", "0", ",", SubscriptBox["nn", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "m"], ",", "0", ",", SubscriptBox["nn", "m"]]], "}"]]]], "]"]], "/;", RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["n", "k"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
