Contribution 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:
with the restrictions k=+-1,+-2,..., abs(a)<1 is still positive
on 0<=x<infinity and has moments a(n).