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 $H_{\lambda}$.

Example. If $\lambda=(4,3,1,1)$ then $H_{\lambda}=(7,4,3,1,5,2,1,2,1)$. Also, $o(\lambda)=6$ 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: \begin{align*} F_1(q,t)&=q, \qquad \qquad F_2(q,t)=2qt, \qquad F_3(q,t)=q^3+2q^2t, \\ F_4(q,t)&=5q^2t^2, \qquad F_5(q,t)=2q^4t+5q^3t^2. \end{align*}

Remark. Observe that $\sum_{n\geq0}F_n(q,q)\,x^n=\prod_{i\geq1}\frac1{1-q^ix^i}$.

Remark. The coefficient of $q^nt^n$ in $F_{2n}(q,t)$ is the convolution of $\sum_{k=0}^np(k)p(n-k)$ where $p(n)$ are the partitions numbers.