Alex Raichev and I have just submitted a journal article on this topic (read it on arXiv). The associated Maple 11 worksheets can be found on Alex’s site. We improve over previous work by Robin Pemantle and me, and derive explicit formulae for the entire asymptotic series obtained by approximating the coefficients of a generating function near a smooth point of the singular variety. The numerical results obtained show how well these approximations work in practice.
I heard a talk by Wojtek Szpankowski a few years ago where he quoted Ziv as saying that second order asymptotics were needed in information theory. Persi Diaconis told me years ago that the work with Robin is all very nice, but what about looking at approximations that work for moderately small values, not just asymptotic approximations. I hope that this paper will satisfy both needs.