New congruences modulo 5 and 9 for partitions with odd parts distinct

Abstract

Let pod(n) denote the number of partitions of an integer n wherein the odd parts are distinct. Recently, a number of congruences for pod(n) have been established. In this paper, we establish the generating function of pod(5n + 2) and then prove new infinite families of congruences modulo 5 and 9 for pod(n) by using the formulas for t ₃(n) and t ₅(n), where t_k (n) is the number of representations of n as a sum of k triangular numbers. In particular, we generalize a congruence for pod(n) due to Radu and Sellers.