According to the pentagonal number theorem:
$\prod_{n=1}^{\infty} (1-q^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}q^{k(3k-1)/2}$
Now the reciprocal of this has the partition numbers $p(k)$ in its power series representation:
$\prod_{n=1}^{\infty} \frac{1}{1-q^{n}} = \sum_{k=0}^{\infty} p(k)q^{k}$
What strikes me about this pair is that there is information about the partitions encoded in the generalized pentagonal numbers.
Are there other pairs of $q$ products and reciprocals for which there is a lot of curiosity about the reciprocal because some number-theoretic information is encoded in the coefficients of the power series representation, and the non-reciprocated power series representation has fairly straightforward exponents?
If the answer to 1 is yes, then also: how do those exponents relate to the information being encoded by the coefficients of the power series representation of the reciprocal?