Does anyone know how to estimate the sums of $p$-powers of $s$-multinomial coefficients, with $p,s\in\mathbb N$ arbitrary? Here's some data that I gathered from OEIS, $$\sum_{a_1+\ldots+a_s=n}\binom{n}{a_1,\ldots,a_s}^p \simeq s^{pn}\sqrt{\frac{K_{sp}}{(\pi n)^{(s-1)(p-1)}}}$$ where $K_{1p}=K_{s1}=1$, $K_{2p}=2^{p-1}/p$, $K_{3p}=1,\frac{27}{16},\frac{81}{16},16,\ldots$ and $K_{s2}=1,1,\frac{27}{16},4,\ldots$

My questions are: (1) is the above formula correct? (2) what's $K_{sp}$, in general?

Many thanks.