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 $n/k \rightarrow \beta > 0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k \Psi^k$. We solve the asymptotically optimal value of $\Psi$ as a function of $p$ and $\beta$.