OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..160
FORMULA
G.f.: sum(a(n)*x^n/(n!)^3,n=0..infinity) = 4*EllipticK((1/2)*sqrt(2-2*sqrt(1-8*x)))^2/Pi^2, sum(a(n)*x^n/(n!)^4,n=0..infinity)=hypergeom([1/2,1/2,1/2],[1,1,1],8*x).
Asymptotics: a(n) = (2*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3)-(1/4)*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3*n)+(1/64)*sqrt(2)/((exp(-1/2))^3*(exp(1/2))^3*n^2)+O(1/n^3))*(2^n)^3/(((1/n)^n)^3*(exp(n))^3), n->infinity.
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*MeijerG([[],[]],[[ -1/2,-1/2,-1/2],[]],x/8)/(8*(Pi)^(3/2)), x=0..infinity), n=0,1... .
This solution is not unique.
a(n) -(2*n-1)^3*a(n-1) +a(n-2) -(2*n-5)^3*a(n-3) =0. - R. J. Mathar, Jul 24 2012
MAPLE
seq((doublefactorial(2*n-1))^3, n=0..15);
MATHEMATICA
Table[((2 n - 1)!!)^3, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)
PROG
(Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^3: n in [0..20]]; // Vincenzo Librandi, Jul 21 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Oct 21 2009
STATUS
approved