Sum[\[Ellipsis] Sum[UnitStep[n - (Subscript[k, 0] + Subscript[k, 1] + Subscript[k, 2] + \[Ellipsis] + Subscript[k, r])]/ (Subscript[k, 0] Subscript[k, 1] Subscript[k, 2] \[Ellipsis] Subscript[k, r]), {Subscript[k, r], 1, n}], {Subscript[k, 0], 1, n}, {Subscript[k, 1], 1, n}, {Subscript[k, 2], 1, n}] == (-((-1)^(r + n)/n!)) Derivative[n][Function[t, Log[t]^(r + 1)/t]][1] /; Element[n, Integers] && n >= 1 && Element[r, Integers] && r >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "0"], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "1"]], "n"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "r"], "=", "1"]], "n"], FractionBox[RowBox[List["UnitStep", "[", RowBox[List["n", "-", RowBox[List["(", RowBox[List[SubscriptBox["k", "0"], "+", SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["k", "r"]]], ")"]]]], "]"]], RowBox[List[SubscriptBox["k", "0"], SubscriptBox["k", "1"], SubscriptBox["k", "2"], "\[Ellipsis]", " ", SubscriptBox["k", "r"]]]]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "n"]]], RowBox[List["n", "!"]]]]], RowBox[List[SuperscriptBox[RowBox[List["Function", "[", RowBox[List["t", ",", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "t", "]"]], RowBox[List["r", "+", "1"]]], "t"]]], "]"]], TagBox[RowBox[List["(", "n", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mrow> <mo minsize='3'> &#8230; </mo> <mo> &#8290; </mo> <mstyle minsize='3'> <mtext> </mtext> </mstyle> <mrow> <mstyle minsize='3'> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> r </mi> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> </mstyle> <mfrac> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> k </mi> <mi> r </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mo> &#8230; </mo> <mo> &#8290; </mo> <msub> <mi> k </mi> <mi> r </mi> </msub> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> r </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mi> t </mi> </mfrac> <mo> ) </mo> </mrow> <semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;n&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> </mrow> </mrow> <msub> <mo> | </mo> <mrow> <mi> t </mi> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> r </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </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> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 0 </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </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> k </ms> <ms> r </ms> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> InterpretationBox </ci> <ms> &#952; </ms> <ci> UnitStep </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 0 </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </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> r </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 0 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> &#8230; </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> r </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> r </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ! </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> log </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <ms> t </ms> <ms> ) </ms> </list> </apply> <ms> t </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> n </ms> <ms> ) </ms> </list> </apply> <ci> Derivative </ci> </apply> </apply> </list> </apply> </list> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> | </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> t </ms> <ms> &#10869; </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </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> r </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "0"], "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "1"], "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "2"], "=", "1"]], "n_"], RowBox[List["\[Ellipsis]_", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k_", "r_"], "=", "1"]], "n_"], FractionBox[RowBox[List["UnitStep", "[", RowBox[List["n_", "-", RowBox[List["(", RowBox[List[SubscriptBox["k_", "0"], "+", SubscriptBox["k_", "1"], "+", SubscriptBox["k_", "2"], "+", "\[Ellipsis]_", "+", SubscriptBox["k_", "r_"]]], ")"]]]], "]"]], RowBox[List[SubscriptBox["k_", "0"], " ", SubscriptBox["k_", "1"], " ", SubscriptBox["k_", "2"], " ", "\[Ellipsis]_", " ", SubscriptBox["k_", "r_"]]]]]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "n"]]], " ", RowBox[List[SuperscriptBox[RowBox[List["Function", "[", RowBox[List["t", ",", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "t", "]"]], RowBox[List["r", "+", "1"]]], "t"]]], "]"]], TagBox[RowBox[List["(", "n", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]]]], RowBox[List["n", "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "0"]]]]]]]]]]

2002-12-18