Some congruences modulo 3, 5 and 7 for broken ((rℓ − 1)/2)-diamond partitions for r ∈ {3,25,49} and odd positive integer ℓ

For any non negative integer n and a fixed positive integer k, let Δk(n) denote the number of broken k-diamond partitions of n. In this paper, we prove some congruences modulo 3 for Δ(3ℓ−1)/2(n), modulo 5 for Δ(25ℓ−1)/2(n), and modulo 7 for Δ(49ℓ−1)/2(n), for odd positive integer ℓ. We also prove a new congruence modulo 5 for Δ2(n).