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

[Max Alekseyev]

On 8/18/07, Paul D. Hanna <pauldhanna at juno.com> wrote:

> 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?)

That follows from the general formula for o.g.f.
I've verified the identity in maple.

Max