Let $\lambda\vdash n$ denote an integer partition of $n$ and $H_{\lambda}$ be the multiset of [hook lengths][1] 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*}

[1]: https://en.wikipedia.org/wiki/Hook_length_formula