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%I #12 Jun 19 2018 03:28:39
%S 1,24,10080,15966720,62270208000,482718652416000,6528287055273984000,
%T 141011000393918054400000,4563680016748763912601600000,
%U 210842016773792892762193920000000
%N a(n) = (4*n)!*(n+1)!/(2*n)!.
%H Harry J. Smith, <a href="/A065236/b065236.txt">Table of n, a(n) for n = 0..50</a>
%F Representation as n-th moment of a positive function (expressible in terms of the hypergeometric functions of type 0F2) on a positive half-axis, in Maple notation: a(n)=int(x^n * (1/6720 * (1260 * hypergeom([], [1/2, -3/4], -1/64 * x) * GAMMA(3/4)^2+4 * x^(7/4) * hypergeom([], [9/4, 11/4], -1/64 * x) * sqrt(Pi) * GAMMA(3/4)-105 * sqrt(2) * Pi * hypergeom([], [ -1/4, 3/2], -1/64 * x) * sqrt(x))/Pi^(1/2)/x^(3/4)/GAMMA(3/4)), x=0..infinity), n=0, 1...
%F Hypergeometric generating function: sum((4 * i)! * (i + 1)! * x^i/((i!)^3 * (2 * i)!), i = 0..infinity) = -1/2 * (-EllipticK(1/2 * sqrt(2) * sqrt((sqrt(1-64 * x)-1)/(1-64 * x)^(1/2))) + 3 * sqrt(1-64 * x) * EllipticK(1/2 * sqrt(2) * sqrt((sqrt(1-64 * x)-1)/(1-64 * x)^(1/2))) + 2 * EllipticE(1/2 * sqrt(2) * sqrt((sqrt(1-64 * x)-1)/(1-64 * x)^(1/2)))) * (1-64 * x)^(1/4)/(64 * x-1)/Pi
%t Table[((4n)!(n+1)!)/(2n)!,{n,0,10}] (* _Harvey P. Dale_, Jan 26 2014 *)
%o (PARI) { for (n=0, 50, write("b065236.txt", n, " ", (4*n)!*(n + 1)!/(2*n)!) ) } \\ _Harry J. Smith_, Oct 14 2009
%K nonn
%O 0,2
%A _Karol A. Penson_, Oct 23 2001
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