Let $a_n$ be a sequence of non-negative real numbers, and $A(x) = \sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ its exponential generating function. Also, suppose $B(x) = \sum_{n=0}^{\infty} b_n \frac{x^n}{n!}$ is such that, for some $k>0$, $[B(x)]^k = A(x)$. If an explicit formula exists for the $a_n$ (or just a formula for the asymptotic behavior), what can be derived about the asymptotic behavior of $b_n$, given $k$?

I'm also interested in the analogous problem for ordinary generating functions.