In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer n with even crank and those with odd crank, denoted \(C(n) = c_e(n) - c_o(n).\) Inspired by Ramanujan’s classical congruences for the partition function p(n), we establish a Ramanujan-type congruence for C(n), proving that \(C(5n+4) \equiv 0 \pmod {5}.\) Further, we study the generating function \(\sum _{n=0}^\infty a(n)\, q^n = \frac{(-q; q)^2_\infty }{(q; q)_\infty },\) which arises naturally in this context, and provide multiple combinatorial interpretations for the sequence a(n). We then offer a complete characterization of the values \(a(n) \mod 2^m\) for \(m = 1, 2, 3, 4,\) highlighting their connection to generalized pentagonal numbers. Using computational methods and modular forms, we also derive new identities and congruences, including \(a(7n+2) \equiv 0 \pmod {7},\) expanding the scope of partition congruences in arithmetic progressions. These results build upon classical techniques and recent computational advances, revealing deep combinatorial and modular structure within partition functions.
