Polynomial analogues of Ramanujan congruences for Han's hooklength formula (Q2851128)

Define the polynomial \(p_n(b)\) by NEWLINE\[NEWLINE \prod_{k \geq 1} (1-q^k)^{b-1} = \sum_{n \geq 0} \frac{q^n}{n!}p_n(b). NEWLINE\]NEWLINE The author studies the (integral) coefficients of \(p_n(b)\pmod 5\) when \(n = 5k+4\). For example, he shows that the coefficients of terms of degree at most \(k\) are all \(0\) modulo \(5\) and that the coefficient of \(b^{k+1+4m}\) is congruent to \(2(-1)^m\binom{k}{m}\pmod 5\). He ends with a discussion of primes other than \(5\) and some open questions.