When one has a question regarding the coefficients of polynomial or power series with integer coefficients, especially if they are positive, it seems like a good strategy to realise the coefficients as counting something, or as being the (graded) dimension of some vector spaces, or to find some other "interpretation".
This often seems to be a difficult task, eg, finding the meaning of the coefficients in the power series expansion of the j-invariant.
Another example is the unimodality of the coefficients of $\prod_{i=1}^n(1+x^i)$ , which can be proven by realising this polynomial as a sum of characters of $\mathfrak{sl_2}$.
Does one have any general strategy to employ when aiming to find an "interpretation" of integer coefficients?
Are there any instances of naturally occurring power series/polynomials with positive integer coefficients in which no "interpretation" is known for their coefficients, and where no interpretation is suspected to exist?
Here we take "interpretation" to mean recognising coefficients as directly counting things, counting things with signs/weights, sums of other polynomials we understand(eg, characters), (graded) dimensions of a chain complex, etc