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*((1/16*(2*Pi^(3/2)*sqrt(2)*hypergeom([], [1/2, 3/4], -(1/256)*x)*sqrt(x) -2*Pi*sqrt(2)*hypergeom([], [3/4, 5/4], -(1/256)*x)*GAMMA(3/4)*x^(3/4) +sqrt(Pi)*GAMMA(3/4)^2*hypergeom([], [5/4, 3/2], -(1/256)*x)*x))*sqrt(2)/(GAMMA(3/4)*x^(5/4)*Pi^(3/2))), x=0..infinity), n=0,1... .
This solution may not be unique.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..100
FORMULA
G.f.: sum(a(n)*x^n/(n!)^3,n=0..infinity)=hypergeom([1/4, 1/2, 3/4], [1, 1], 256*x).
a(n) ~ 2*(1-1/(16*n)+1/(512*n^2)+331/(122880*n^3)+O(1/n^4)))*(2^n)^8/(((1/n)^n)^3*(exp(n))^3),n->infinity.
1/a(n) = n!*[x^n](cosh(x^(1/4))+cos(x^(1/4)))/2. - Peter Luschny, Jul 12 2012
MAPLE
A166338_list := proc(n) u:=z^(1/4); (cosh(u)+cos(u))/2:series(%, z, n+2):
seq(1/(i!*coeff(%, z, i)), i=0..n) end: A166338_list(9); # Peter Luschny, Jul 12 2012
MATHEMATICA
Table[(4n)!/n!, {n, 0, 10}] (* Harvey P. Dale, May 30 2015 *)
PROG
(Magma) [Factorial(4*n) / Factorial(n): n in [0..15]]; // Vincenzo Librandi, May 10 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Oct 12 2009
STATUS
approved