Formula for A120014(n), Anyone?

[Max]

On 6/7/06, Paul D. Hanna <pauldhanna at juno.com> wrote:
> Seqfans,
>      Perhaps someone can find a formula for a(n) of the following
> sequence?
> It is not yet visible in OEIS, so I copy the entry below.
> The best I can do so far is:
>
> a(n) = [x^n] x*((1-n+n^2) - n^2*(n+1)*x - n*(1-(n+2)*x)*C(x)
> )/(1-n+n^2*x)^2,
> where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
>
> I believe that there is a nice closed form for a(n), but it escapes me
> ...

It follows directly from your formula that

a(n) = (n^2/(n-1))^(n+1) / n^4 * ( 2*n-1  + \sum_{k=0}^{n-1}
(2*(k+1)/n - k - 3) * ((n-1)/n^2)^k * C_k )

where C_k is the k-th Catalan number.

But this formula still leaves two questions:
1. How to see (directly from the formula) that a(n) is integer?
2. How to prove that a(n) is divisible by n.

Max