Approximating Stirling's number of the second kind to allow for large inputs

Question

I'm looking for an approximation for Stirling's number of the second kind, $S_2(n,k)$, which counts the ways to partition a set of $n$ objects into $k$ non-empty subsets:

( http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Simple_identities )

I need to compute $S_2(n,k)$ for values of $n$ up to $10^6$ and values of $k$ up to $10^5$ or so. Is this possible? Can we bound an error term for the approximation or at least say that the approximation becomes more accurate as $n \to \infty$ and/or $k \to \infty$?

Answer 1

See the asymptotic approximations 26.8.42 and 26.8.43 here. That page also gives further references. See also this article and this slide show.