Infinite families of congruences modulo 7 for broken 3-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu conjectured that . Jameson confirmed this conjecture and proved that by using the theory of modular forms. In this paper, we prove several infinite families of Ramanujan-type congruences modulo 7 for by establishing a recurrence relation for a sequence related to . In the process, we also give new proofs of the four congruences due to Paule and Radu, and Jameson.