On the number of partitions with distinct even parts (Q536218)

Let \(ped(n)\) be defined as \[ \sum_{n=0}^\infty ped(n)q^n = \prod_{n=1}^\infty \frac{(1-q^{4n})}{(1-q^n)}. \] In this article, the author used the theory of modular forms to establish various congruences satisfied by \(ped(n)\). For example, the author showed that if \(\tau(n)\) is the Ramanujan \(\tau\) function and \(\tau(\ell)\equiv 0\pmod{2},\) then for integers \(n\geq 0\) and \(\alpha\geq 1\), \[ ped\left(\ell^{2\alpha+1}n +\frac{s\ell^{2\alpha}-1}{8}\right) \equiv 0\pmod{2}. \] As a special case, the author deduced that \[ ped\left(5^{2\alpha+1}n+\frac{17\cdot 5^{2\alpha}-1}{8}\right)\equiv 0\pmod{2}. \]