Formula for A120014(n) -- Simplified

[Max]

On 6/8/06, Paul D. Hanna <pauldhanna at juno.com> wrote:

>     Using the lovely formula for the n-th term of the m-th power of
> Catalan function:
> [x^n] Catalan(x)^m = m*C(2*n+m-1,n)/(n+m),
> I have found the simplest form yet for A120014:
>
> a(n) = n^(n-1) - Sum_{k=1..n-1}
> n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!/n!

I've slightly different and somewhat simpler formula:

a(n) = Sum_{k=1..n} n^(k-3)*(n-k+1)*k*C(2*n-k-1,n-1)

implying that a(n) is the coefficient of x^n in the expansion of

(x*(1-x+x*n)) / (n^2 * (1-n*x)^2 * (1-x)^(n+1))

> Thus, the e.g.f. of A120014 does involve the LambertW function as
> suspected

It still does not sound to me as a strong evidence.

> (I am still working on obtaining that e.g.f.).

Any luck so far?

Max