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variants of this functions
Factorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Summation > Infinite summation > Parameter-containing sums





http://functions.wolfram.com/06.01.23.0058.01









  


  










Input Form





Sum[(4 k)!/((4 k + 3 n + 2)! 9^k), {k, 0, Infinity}] == ((1 - Sqrt[3])^(3 n + 1) (Sqrt[3] Log[Sqrt[3]/(Sqrt[3] - 1)]))/ (4 (3 n + 1)!) + (1/4) Sum[(Sqrt[3] (Sqrt[3] + 1)^j - (1 - Sqrt[3])^j Sqrt[3])/ ((-3 n - 1 + j) (3 n + 1)!), {j, 0, 3 n}] + ((Sqrt[3] + 1)^(3 n + 1) (Sqrt[3] Log[(Sqrt[3] + 1)/Sqrt[3]]))/ (4 (3 n + 1)!) + (1/2) (Sum[((-8)^j (-1 + 9 n - 9 j))/((-n + j) (1 - 3 n + 3 j) (3 n + 1)!), {j, 0, n - 2}] - ((3 (-8)^n) (-(Pi/(6 Sqrt[3])) + (1/6) (1 - 3 Log[3] + 3 Log[4])))/ (3 n + 1)!) /; Element[n, Integers] && n >= 1










Standard Form





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MathML Form







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</mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msqrt> <mn> 3 </mn> </msqrt> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </munderover> <mfrac> <mrow> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> 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</apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k_"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k_"]], "+", RowBox[List["3", " ", "n_"]], "+", "2"]], ")"]], "!"]], " ", SuperscriptBox["9", "k_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[SqrtBox["3"], RowBox[List[SqrtBox["3"], "-", "1"]]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["3", " ", "n"]]], FractionBox[RowBox[List[RowBox[List[SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], "j"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox["3"]]], ")"]], "j"], " ", SqrtBox["3"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "n"]], "-", "1", "+", "j"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox["3"], "+", "1"]], ")"]], RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]]], " ", RowBox[List["(", RowBox[List[SqrtBox["3"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List[SqrtBox["3"], "+", "1"]], SqrtBox["3"]], "]"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["9", " ", "n"]], "-", RowBox[List["9", " ", "j"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "j"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "n"]], "+", RowBox[List["3", " ", "j"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "8"]], ")"]], "n"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", RowBox[List["6", " ", SqrtBox["3"]]]]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", RowBox[List["Log", "[", "3", "]"]]]], "+", RowBox[List["3", " ", RowBox[List["Log", "[", "4", "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n"]], "+", "1"]], ")"]], "!"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]]]










Contributed by





Troy Kessler










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





2007-05-02





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