I've encountered the following sum: $$ s_n = \sum_{j=1}^n {n \brace j}(\alpha n)_j \beta^n. $$ Here $\alpha$ and $\beta$ are positive constants, $(\alpha n)_j$ is a falling power, and ${n \brace j}$ is the Stirling number of second kind. My computations suggests that $s_n$ grows like $n^n C^n$ (where the constant $C$ depends on $\alpha$ and $\beta$); specifically, $\sqrt[n]{(s_n/n^n)}$ seems to converge to a limit as $n \rightarrow \infty$.

Are there good approaches to figuring out these kinds of limits? A similar situation was discussed in this question, but the sum there had $(\alpha)_j$ rather than $(\alpha n)_j$; the extra dependence on $n$ is causing me headaches!