[seqfan] Inversion Formula for Peculiar G.F. - Help

[Paul D Hanna]

What is the non-recursive inversion of the equation:
 
F(x) = (1-x) + a(1)*x*(1-x)(1-2x) + a(2)*x^2*(1-x)(1-2x)(1-3x) + ...
             + a(n)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)x) + ...
 
Knowing F(x), can one determine a(n) without recursion?
 
There is a recurrence formula for a(n) using Stirling1,
but I want an explicit form for the g.f. A(x) 
where
A(x) = Sum_{n>=0} a(n)*x^n.
 
EXAMPLE.
I found that sequence A082161 has such a g.f.: 
http://www.research.att.com/projects/OEIS?Anum=A082161
1,3,16,127,1363,18628,311250,6173791,142190703,3737431895,
 
where
 
1 = (1-x) + x*(1-x)*(1-2*x) + 3*x^2*(1-x)*(1-2*x)*(1-3*x) +
    16*x^3*(1-x)*(1-2*x)*(1-3*x)*(1-4*x) +
  127*x^4*(1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x) +
1363*x^5*(1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x) + ...
 
and from this I would like to determine A082161(n)
or its g.f.: A(x) explicitly. 
 
Any comments would be welcome.
Thanks,
     Paul