How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

Question

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ such pairs, and for $n=3$ there are $q^8+q^7+q^6-q^5-q^4$ such pairs. Can anybody recognize these polynomials, generalize to arbitrary $n$, and prove the result?