BoxData[\(\(\(Sum[\(\(\(Multinomial[\(\(p - \(Sum[\(\(Subscript[\(j, n\)]\), \({h, 1, n}\)\)]\)\), \(Subscript[\(j, 1\)]\), \(Subscript[\(j, 2\)]\), …, \(Subscript[\(j, n\)]\)\)]\) * \(Multinomial[\(\(q + \(Sum[\(\(Subscript[\(j, n\)]\), \({h, 1, n}\)\)]\)\), \(\(Subscript[\(k, 1\)]\) - \(Subscript[\(j, 1\)]\)\), \(\(Subscript[\(k, 2\)]\) - \(Subscript[\(j, 2\)]\)\), …, \(\(Subscript[\(k, n\)]\) - \(Subscript[\(j, n\)]\)\)\)]\)\), \({\(Subscript[\(j, 1\)]\), 0, \(Subscript[\(k, 1\)]\)}\), \({\(Subscript[\(j, 2\)]\), 0, \(Subscript[\(k, 2\)]\)}\), …\ ,\(\ \ \)\({\(Subscript[\(j, n\)]\), 0, \(Subscript[\(k, n\)]\)}\)\)]\)  \(Multinomial[\(\(p + q\), \(Subscript[\(k, 1\)]\), \(Subscript[\(k, 2\)]\), …, \(Subscript[\(k, n\)]\)\)]\)\)/;\(\(\(Subscript[\(k, 1\)]\) ∈ Integers\) && \(\(Subscript[\(k, 2\)]\) ∈ Integers\) && … && \(\(Subscript[\(k, n\)]\) ∈ Integers\) && \(\(Subscript[\(k, 1\)]\) ≥ 0\) && \(\(Subscript[\(k, 2\)]\) ≥ 0\) && … && \(\(Subscript[\(k, n\)]\) ≥ 0\) && \(p ∈ Integers\) && \(p ≥ 0\) && \(q ∈ Integers\) && \(q ≥ 0\) && \(n ∈ Integers\) && \(n > 0\)\)\)]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], SubscriptBox["k", "1"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], SubscriptBox["k", "2"]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", SubscriptBox["j", RowBox[List["n", "=", "0"]]], SubscriptBox["k", "n"]], RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["p", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "n"], SubscriptBox["j", "n"]]]]], ",", SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["j", "n"]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["q", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "1"]], "n"], SubscriptBox["j", "n"]]]]], ",", RowBox[List[SubscriptBox["k", "1"], "-", SubscriptBox["j", "1"]]], ",", RowBox[List[SubscriptBox["k", "2"], "-", SubscriptBox["j", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["k", "n"], "-", SubscriptBox["j", "n"]]]]], "]"]]]]]]]]]]]], "\[Equal]", " ", RowBox[List["Multinomial", "[", RowBox[List[RowBox[List["p", "+", "q"]], ",", SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["k", "n"]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["k", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[Element]", "Integers"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["k", "n"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["k", "1"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["k", "2"], "\[GreaterEqual]", "0"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["k", "n"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]

<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> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> p </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> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> q </mi> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> k </mi> <mi> h </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> k </mi> <mi> n </mi> </msub> <mo> - </mo> <msub> <mi> j </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> k </mi> <mi> h </mi> </msub> </mrow> </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> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <msub> <mi> k </mi> <mi> n </mi> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8230; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> h </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> h </ms> </apply> </list> </apply> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 1 </ms> </apply> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> n </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> + </ms> <ms> q </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> h </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> h </ms> </apply> </list> </apply> </list> </apply> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> n </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <ms> &#8230; </ms> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> n </ms> </apply> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#8469; </ms> <ms> + </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

2002-12-18