G.f. for a(n) = C(2n,n)*Sum_{k=0..2n} T(n,k)/C(2n,k) ?

[Paul D. Hanna]

Seqfans, 
      In reply to my own question: 
> Can anyone provide a general formula (by inspection) for the g.f. of: 
> (1) a(n) = C(2n,n) * Sum_{k=0..2n} T(n,k) / C(2n,k) 
> where T(n,k) = [x^k] (1 + b*x + c*x^2)^n. 
EXAMPLE; case b=c=1: 
> a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5 ; 
> a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21 ; 
> a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20)
= 83. 
...
> G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3). 
  
By inspection, the formula for the g.f. of (1), when c=1 is: 
 
(2) G.f.: A(x) = 1/sqrt(1 + d*x + e*x^2 + f*x^3)  where 
d = -2*b - 8 ;
e =  b^2 + 12*b + 20 ;
f = -4*b^2 - 16*b - 16 ;
(proof, anyone?) 
but there does not appear to be such a simple formula for |c| > 1. 
 
Yet is there a nice g.f. for (1) when |c| > 1, |b| > 0  ? 
  
Paul