OFFSET
0,2
COMMENTS
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n) = int(x^n*(1/3)*sqrt(2)*BesselK(1/3,(2/9)*sqrt(6*x))/(sqrt(x)*Pi), x=0..infinity), n=0,1... .
This solution is unique.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..100
FORMULA
G.f.: sum(a(n)*x^(n)/(n!)^2,n=0..infinity)=hypergeom([1/3, 2/3], [1], (27/2)*x).
Asymptotics: a(n)=(sqrt(3)-(1/18)*sqrt(3)/n+(1/648)*sqrt(3)/n^2 +(463/174960)*sqrt(3)/n^3+O(1/n^4))*(3^n)^3/(((1/n)^n)^2*(exp(n))^2*2^n), n->infinity.
E.g.f.: (of aerated sequence) 2*sqrt(2)*cos(arcsin((3*sqrt(6)x/4)/3))/sqrt(8-27x^2). - Paul Barry, Jul 27 2010
2*a(n) = 3*(3*n-1)*(3*n-2)*a(n-1). - R. J. Mathar, Jul 24 2012
MATHEMATICA
Table[(3*n)!/(2^n*n!), {n, 0, 10}] (* G. C. Greubel, May 09 2016 *)
PROG
(Magma) [Factorial(3*n)/(2^n*Factorial(n)): n in [0..20]]; // Vincenzo Librandi, May 10 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Oct 12 2009
STATUS
approved