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%I #9 Jun 26 2022 10:50:35
%S 1,10,280,12600,776160,60540480,5708102400,630745315200,
%T 79894406592000,11408921261337600,1812981305892556800,
%U 317271728531197440000,60623305667038033920000,12557684745315021312000000
%N a(n)=(2*n+1)!*(2*n-2)!/((n-1)!*(n!)^2*6) ,n=1,2... .
%C Representation of a(n) as n-th moment of a positive weight function on a positive half-axis, in Maple notation: a(n)=int(x^n*(1/(48*(Pi)^(3/2)))*exp(-x/32)*BesselK(1,x/32)/sqrt(x), x=0..infinity), n=1,2... .
%F E.g.f.: (1/12)*(Pi+2*EllipticK(4*sqrt(x))-4*EllipticE(4*sqrt(x)))/Pi
%F Conjecture: n*a(n) -4*(2*n+1)*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Jun 08 2016
%F a(n) ~ 2^(4*n - 3/2) * n^(n - 1/2) / (3 * sqrt(Pi) * exp(n)). - _Vaclav Kotesovec_, Jun 26 2022
%o (PARI) a(n)=(2*n+1)!*(2*n-2)!/((n-1)!*(n!)^2*6); \\ _Michel Marcus_, Aug 17 2013
%K nonn
%O 1,2
%A _Karol A. Penson_, Mar 04 2009