We study infinite families of generating functions involving the rank of the ordinary partition function, which include as special cases many of the generating functions introduced by Atkin and Swinnerton-Dyer in the 1950s. We prove that each of these generating functions is a weakly holomorphic modular form of weight 1/2 on some congruence subgroup Γ1(N). The results depend on linear relations among the non-holomorphic parts of a family of weak Maass forms recently introduced by Bringmann and Ono. Dedicated to Professor Wolfgang Schmidt on the occasion of his seventy-fifth birthday.