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 Factorial

 http://functions.wolfram.com/06.01.23.0049.01

 Input Form

 Sum[(2 k)!^2/(2 k + 2 n)!^2, {k, 0, Infinity}] == (1/2) Sum[-((3 (4 n - 2)! (2 n - j - 2)!^2)/((2 n - 1)!^4 (4 n - 2 j - 2)!)), {j, 0, 2 n - 2}] + ((4 n - 2)! Pi^2)/(12 (2 n - 1)!^4) + (1/2) (Sum[((-16)^j (4 - 5 n + 5 j) (-1 + n)!^2 (-1 + 2 n - 2 j)!)/ ((1 - n + j) (1 - 2 n + 2 j)^2 (-1 + 2 n)!^3 (-1 + n - j)!^2), {j, 0, n - 2}] + ((n!^2 -1 (-16)^n) (3 - 4 Log[2]))/ (4 (2 n - 1)! (2 n)!^2)) /; Element[n, Integers] && n >= 1

 Standard Form

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 MathML Form

 k = 0 ( ( 2 k ) ! ) 2 ( ( 2 k + 2 n ) ! ) 2 π 2 ( 4 n - 2 ) ! 12 ( ( 2 n - 1 ) ! ) 4 + 1 2 ( ( ( n ! ) 2 ( - 1 ) ( - 16 ) n ) ( 3 - 4 log ( 2 ) ) 4 ( 2 n - 1 ) ! ( ( 2 n ) ! ) 2 + j = 0 n - 2 ( - 16 ) j ( 5 j - 5 n + 4 ) ( ( n - 1 ) ! ) 2 ( - 2 j + 2 n - 1 ) ! ( j - n + 1 ) ( 2 j - 2 n + 1 ) 2 ( ( 2 n - 1 ) ! ) 3 ( ( - j + n - 1 ) ! ) 2 ) + 1 2 j = 0 2 n - 2 - 3 ( 4 n - 2 ) ! ( ( - j + 2 n - 2 ) ! ) 2 ( ( 2 n - 1 ) ! ) 4 ( - 2 j + 4 n - 2 ) ! /; n + Condition k 0 2 k 2 2 k 2 n 2 -1 2 4 n -2 12 2 n -1 4 -1 1 2 n 2 -1 -16 n 3 -1 4 2 4 2 n -1 2 n 2 -1 j 0 n -2 -16 j 5 j -1 5 n 4 n -1 2 -2 j 2 n -1 j -1 n 1 2 j -1 2 n 1 2 2 n -1 3 -1 j n -1 2 -1 1 2 j 0 2 n -2 -1 3 4 n -2 -1 j 2 n -2 2 2 n -1 4 -2 j 4 n -2 -1 n SuperPlus [/itex]

 Rule Form

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 Contributed by

 Troy Kessler

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

 2007-05-02