A161149 a(n) = (2*n)!*(2*n+1)!/n! = n!*A000909(n), n=0,1...

1, 12, 1440, 604800, 609638400, 1207084032000, 4142712397824000, 22619209692119040000, 184572751087691366400000, 2146211949647675208499200000, 34253542716376896327647232000000, 727956289808441800755158974464000000, 20091593598712993700842387695206400000000

Offset: 0

Author

Karol A. Penson, Jun 03 2009

Keywords

• nonn

Comments

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... .

Explicit form of the function W(x) is

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);

This is the solution of the Stieltjes moment problem with the moments a(n).

This solution may not be unique.

Crossrefs

Cf. A000909, A000165.

Programs

Magma

[Factorial(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 06 2015

Formula

Hypergeometric generating function: sum(a(n)*x^n/(n!)^4, n=0..infinity)= -2*EllipticE(4*sqrt(x))/((16*x-1)*Pi).