Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series

In their epic paper [176], [242, pp. 276–309], G.H. Hardy and S. Ramanujan found an asymptotic formula for the partition function p(n) that arises from the power series coefficients of the reciprocal of the Dedekind eta function. As they indicated near the end of their paper, their methods also apply to several analogues of the partition function generated by modular forms of negative weight that are analytic in the upper half-plane. In their last published paper [177], [242, pp. 310–321], they considered a similar problem for the coefficients of modular forms of negative weight having a simple pole in a fundamental region, and in particular, they applied their theorem to find interesting series representations for the coefficients of the reciprocal of the Eisenstein series E₆(Γ).