  In his study of Nekrasov-Okounkov type formulas on "partition theoretic"
expressions for families of infinite products, Han discovered seemingly
unrelated $q$-series that are supported on precisely the same terms as these
infinite products. In earlier work with Ono, Han proved one instance of this
occurrence that exhibited a relation between numbers $a(n)$ that are given in
terms of hook lengths of partitions, with numbers $b(n)$ that equal the number
of 3-core partitions of $n$. Recently Han revisited the $q$-series with
coefficients $a(n)$ and $b(n)$, and numerically found a third $q$-series whose
coefficients appear to be supported on the same terms. Here we prove Han's
Conjecture about this third series by proving a general theorem about this
phenomenon.
