On a nonlinear relation for computing the overpartition function

Abstract

In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademachertype convergent series that can be used to compute an isolated value of the overpartition function p(n). Computing p(n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p(n) that requires only the values of p(k) with k n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p(n) to obtain a simple and fast computation of the value of p(n). This new method uses only (large) integer arithmetic and it is simpler to program.