On raw moments of the binomial distribution

We study the $k$:th raw moment of a variable $R$ following the binomial distribution $\text{B}(n,p)$, where $n\mathrm{/}k\to \beta >0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k{\mathrm{\Psi }}^k$. We solve the asymptotically optimal value of $\mathrm{\Psi }$ as a function of $p$ and $\beta$.