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 Multinomial

 http://functions.wolfram.com/06.04.06.0001.01

 Input Form

 Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] \[Proportional] (Subscript[n, 1]^(a - 1)/Product[Gamma[Subscript[n, k] + 1], {k, 2, m}]) Sum[((((-1)^k Pochhammer[1 - a, k])/k!) B[k, a, a])/Subscript[n, 1]^k, {k, 0, Infinity}] /; (Abs[Subscript[n, 1]] -> Infinity) && a == 1 + Sum[Subscript[n, k], {k, 2, m}] && B[n, \[Alpha], z] == n! SeriesTerm[(t^\[Alpha] E^(t z))/(E^t - 1)^\[Alpha], {t, 0, n}] && Abs[Arg[Subscript[n, 1] + a]] < Pi

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SubsuperscriptBox["n", "1", RowBox[List["a", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "k"], "+", "1"]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", "k"]], "]"]], " "]], RowBox[List["k", "!"]]], RowBox[List["B", "[", RowBox[List["k", ",", "a", ",", "a"]], "]"]], " ", SubsuperscriptBox["n", "1", RowBox[List["-", "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["n", "1"], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["a", "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], SubscriptBox["n", "k"]]]]]]], "\[And]", RowBox[List[RowBox[List["B", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["t", "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["t", " ", "z"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "-", "1"]], ")"]], "\[Alpha]"]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List[SubscriptBox["n", "1"], "+", "a"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]

 MathML Form

 ( n 1 + n 2 + + n m ; n 1 , n 2 , , n m ) 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]] n 1 a - 1 k = 2 m Γ ( n k + 1 ) k = 0 ( - 1 ) k ( 1 - a ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "k"], Pochhammer] B ( k , a , a ) n 1 - k k ! /; ( "\[LeftBracketingBar]" n 1 "\[RightBracketingBar]" "\[Rule]" ) a k = 2 m n k + 1 B ( n , α , z ) n ! ( [ t n ] t α t z ( t - 1 ) α ) "\[LeftBracketingBar]" arg ( a + n 1 ) "\[RightBracketingBar]" < π ( n 1 + n 2 + + n m ; n 1 , n 2 , , n m ) 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]] n 1 a - 1 k = 2 m Γ ( n k + 1 ) k = 0 ( - 1 ) k ( 1 - a ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "k"], Pochhammer] B ( k , a , a ) n 1 - k k ! /; ( "\[LeftBracketingBar]" n 1 "\[RightBracketingBar]" "\[Rule]" ) a k = 2 m n k + 1 B ( n , α , z ) n ! ( [ t n ] t α t z ( t - 1 ) α ) "\[LeftBracketingBar]" arg ( a + n 1 ) "\[RightBracketingBar]" < π [/itex]

 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[FractionBox[RowBox[List[SubsuperscriptBox["nn", "1", RowBox[List["a", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["B", "[", RowBox[List["k", ",", "a", ",", "a"]], "]"]], " ", SubsuperscriptBox["nn", "1", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "k"], "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["nn", "1"], "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["a", "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], SubscriptBox["nn", "k"]]]]]]], "&&", RowBox[List[RowBox[List["B", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["t", "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["t", " ", "z"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "t"], "-", "1"]], ")"]], "\[Alpha]"]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List[SubscriptBox["nn", "1"], "+", "a"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]

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

 2001-10-29