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 Factorial

 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

 Cell[BoxData[RowBox[List[RowBox[List[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"]]]]]], "\[Equal]", 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"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]

 MathML Form

 k = 0 ( 4 k ) ! ( 4 k + 3 n + 2 ) ! 9 k ( 3 log ( 3 + 1 3 ) ) ( 3 + 1 ) 3 n + 1 4 ( 3 n + 1 ) ! + 1 2 ( j = 0 n - 2 ( - 8 ) j ( - 9 j + 9 n - 1 ) ( j - n ) ( 3 j - 3 n + 1 ) ( 3 n + 1 ) ! - ( 3 ( - 8 ) n ) ( 1 6 ( - 3 log ( 3 ) + 3 log ( 4 ) + 1 ) - π 6 3 ) ( 3 n + 1 ) ! ) + ( 1 - 3 ) 3 n + 1 ( 3 log ( 3 3 - 1 ) ) 4 ( 3 n + 1 ) ! + 1 4 j = 0 3 n 3 ( 3 + 1 ) j - ( 1 - 3 ) j 3 ( j - 3 n - 1 ) ( 3 n + 1 ) ! /; n + Condition k 0 4 k 4 k 3 n 2 9 k -1 3 1 2 3 1 2 1 3 1 2 -1 3 1 2 1 3 n 1 4 3 n 1 -1 1 2 j 0 n -2 -8 j -9 j 9 n -1 j -1 n 3 j -1 3 n 1 3 n 1 -1 -1 3 -8 n 1 6 -3 3 3 4 1 -1 6 3 1 2 -1 3 n 1 -1 1 -1 3 1 2 3 n 1 3 1 2 3 1 2 3 1 2 -1 -1 4 3 n 1 -1 1 4 j 0 3 n 3 1 2 3 1 2 1 j -1 1 -1 3 1 2 j 3 1 2 j -1 3 n -1 3 n 1 -1 n SuperPlus [/itex]

 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