OFFSET
0,2
COMMENTS
From Karol A. Penson, Jun 11 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) on the positive axis, in Maple notation:
a(n)=int(x^n*W(x),x=0..infinity)=int(x^n*(1/4)*exp(-x^(1/4))/x^(3/4),x=0..infinity), n=0,1... .
This is the solution of the Stieltjes moment problem with the moments a(n), n=0,1... .
As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
This is indeed the case as by construction the following "doubly" infinite family:
V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),
with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
E.g.f.: 1/(1-x^4).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021
MATHEMATICA
(4*Range[0, 20])! (* Harvey P. Dale, Oct 03 2014 *)
PROG
(Magma) [Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Dec 08 2004
EXTENSIONS
More terms from Harvey P. Dale, Oct 03 2014
STATUS
approved