Asymptotic formulas for coefficients of inverse theta functions

We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms. These coefficients can be viewed as a refinement of the numbers $p_k(n)$ of partitions of n into k colors. Part of the motivation for this work is that they are equal to the Betti numbers of the Hilbert scheme of points on an algebraic surface S and appear also as counts of Bogomolny-Prasad-Sommerfield (BPS) states in physics.