This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their *Dyson ranks*.

I would like to ask:
>**QUESTION.** Is this congruence true modulo $2$?
$$\sum_{n\geq0}\frac{q^{\binom{n+1}2}}{\prod_{j=1}^n(1-q^j)}\sum_{m=1}^n\frac{q^m}{1-q^m}\equiv
\sum_{n\geq0}q^{3n+2}\prod_{j=1}^n(1-q^{4j-1}).$$

**Clarification.** As Richard Stanley commented on this, I meant to say that after expanding the two $q$-series, the coefficients agree term-by-term modulo $2$.