I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ for $p,n=1,2,\ldots$.

**Background**

The reason we expect this equality to hold is the following: The left hand side appear in [arXiv:1307.7560] as the moments of the squared singular values of a product of $n$ Gaussian non-Hermitian $N\times N$ matrices, while the right hand side (Fuss-Catalan numbers) is the asymptotic prediction obtained using techniques from free probability, see e.g. [arXiv:0710.5931]. Thus the results should agree at leading order in $N$.