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 CatalanNumber

 http://functions.wolfram.com/06.41.06.0003.01

 Input Form

 CatalanNumber[z] == (1/Sqrt[Pi]) Sum[(j + 1) Subscript[d, k - j] Sum[(((-1)^r Binomial[j, r])/(1 + r)) Subscript[p, r, j] z^k, {r, 0, j}], {k, 0, Infinity}, {j, 0, k}] /; Subscript[a, k] == (1/k!) Derivative[k][Gamma][1/2] && Subscript[b, k] == Log[4]^k/k! && Subscript[c, k] == Derivative[k][Gamma][2]/k! && Subscript[d, k] == Sum[Subscript[a, n] Subscript[b, k - n], {n, 0, k}] && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[c, m] Subscript[p, j, k - m], {m, 1, k}] && Element[k, Integers] && k >= 0 && Abs[z] < 1/2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], SubscriptBox["d", RowBox[List["k", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], RowBox[List["1", "+", "r"]]], SubscriptBox["p", RowBox[List["r", ",", "j"]]], SuperscriptBox["z", "k"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[" ", RowBox[List["k", "!"]]]]], RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", FractionBox["1", "2"], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "4", "]"]], "k"], RowBox[List["k", "!"]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "2", "]"]], RowBox[List[" ", RowBox[List["k", "!"]]]]]]], "\[And]", RowBox[List[SubscriptBox["d", "k"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "n"], SubscriptBox["b", RowBox[List["k", "-", "n"]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["c", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", FractionBox["1", "2"]]]]]]]]]

 MathML Form

 C TagBox["C", CatalanNumber] z 1 π k = 0 j = 0 k ( j + 1 ) d k - j r = 0 j ( - 1 ) r r + 1 ( j r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["r", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] p r , j z k /; a k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( 1 2 ) k ! b k log k ( 4 ) k ! c k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( 2 ) k ! d k n = 0 k a n b k - n p j , 0 1 p j , k 1 k m = 1 k ( - k + j m + m ) c m p j , k - m k "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 2 Condition CatalanNumber z 1 1 2 -1 j 0 k k 0 j 1 Subscript d k -1 j r 0 j -1 r r 1 -1 Binomial j r Subscript p r j z k Subscript a k D Gamma 1 2 1 2 k k -1 Subscript b k 4 k k -1 Subscript c k D Gamma 2 2 k k -1 Subscript d k n 0 k Subscript a n Subscript b k -1 n Subscript p j 0 1 Subscript p j k 1 k -1 m 1 k -1 k j m m Subscript c m Subscript p j k -1 m k z 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", SubscriptBox["d", RowBox[List["k", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["r", ",", "j"]]], " ", SuperscriptBox["z", "k"]]], RowBox[List["1", "+", "r"]]]]]]]]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", FractionBox["1", "2"], "]"]], RowBox[List["k", "!"]]]]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "4", "]"]], "k"], RowBox[List["k", "!"]]]]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "2", "]"]], RowBox[List["k", "!"]]]]], "&&", RowBox[List[SubscriptBox["d", "k"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "n"], " ", SubscriptBox["b", RowBox[List["k", "-", "n"]]]]]]]]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["c", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", FractionBox["1", "2"]]]]]]]]]]]

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

 2007-05-02