Let $\lambda\vdash n$ denote an integer partition of $n$ and $H_{\lambda}$ be the multiset of hook lengths of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even entries in $H_{\lambda}$. For example, if $\lambda=(4,3,1,1)$ then $H_{\lambda}=(7,4,3,1,5,2,1,2,1)$. Also, $o(\lambda)=5$ and $e(\lambda)=3$.

QUESTION. Is there a generating function for the following "parity distribution" function? $$F_n(q,t):=\sum_{\lambda\vdash n}q^{o(\lambda)}t^{e(\lambda)}.$$

Examples. Here is a short list of such polynomials: $F_1(q,t)=q, F_2(q,t)=2qt, F_3(q,t)=q^3+2q^2t, F_4(q,t)=5q^2t^2, F_5(q,t)=2q^4t+5q^3t^2$.