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%I #12 Sep 08 2022 08:45:45
%S 1,12,1440,604800,609638400,1207084032000,4142712397824000,
%T 22619209692119040000,184572751087691366400000,
%U 2146211949647675208499200000,34253542716376896327647232000000,727956289808441800755158974464000000,20091593598712993700842387695206400000000
%N a(n) = (2*n)!*(2*n+1)!/n! = n!*A000909(n), n=0,1...
%C Integral representation as n-th moment of a positive function W(x) expressed in terms of Meijer's G-function on the positive axis, in Maple notation: a(n)= int(x^n*W(x),x=0..infinity)= int(x^n*(1/2)*MeijerG([[], []], [[1, 1/2, 0], []], (1/16)*x) /(sqrt(x)*Pi),x=0..infinity), n=0,1... .
%C Explicit form of the function W(x) is
%C W(x)=1/2*(-1/2*Pi*x^(1/2)*hypergeom([],[1/2, 3/2],-1/16*x)+Pi^(1/2)-1/16*Pi^(1/2)*x*sum((-1)^(2*j)* (-Psi(3/2+j)+Pi*tan(Pi*j)-Psi(2+j)-Psi(1+j)-4*log(2)+log(x))*2^(-4*j)* (x^j)*sec(Pi*j)*2^(2*j)/(1/2+j)/GAMMA(1+2*j)/GAMMA(2+j), j = 0..infinity))/(x^(1/2)*Pi);
%C This is the solution of the Stieltjes moment problem with the moments a(n).
%C This solution may not be unique.
%F Hypergeometric generating function: sum(a(n)*x^n/(n!)^4, n=0..infinity)= -2*EllipticE(4*sqrt(x))/((16*x-1)*Pi).
%o (Magma) [Factorial(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 06 2015
%Y Cf. A000909, A000165.
%K nonn
%O 0,2
%A _Karol A. Penson_, Jun 03 2009
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