A166334 a(n) = (3*n)!/(2^n*n!).

1, 3, 90, 7560, 1247400, 340540200, 138940401600, 79196028912000, 60109785944208000, 58607041295602800000, 71383376298044210400000, 106218463931489785075200000, 189599958117709266359232000000

Offset: 0

Author

Karol A. Penson, Oct 12 2009

Keywords

• nonn

Comments

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem): a(n) = Integral_{x=0..oo} x^n*(1/3)*sqrt(2)*BesselK(1/3,(2/9)*sqrt(6*x))/(sqrt(x)*Pi) dx, n >= 0.

This solution is unique.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..100

Programs

Magma

[Factorial(3*n)/(2^n*Factorial(n)): n in [0..20]]; // Vincenzo Librandi, May 10 2016

Mathematica

Table[(3*n)!/(2^n*n!), {n, 0, 10}] (* G. C. Greubel, May 09 2016 *)

Formula

G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 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