TITLE: Alternating sum of hook lengths: Part I
QUESTION [12 upvotes]: Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$.

Is there a closed formula or a generating function for the following sequence?
$$f_n=\sum_{\lambda\vdash n}\sum_{(i,j)\in\lambda}(-1)^{i+j}\cdot h_{ij}(\lambda)$$

REPLY [3 votes]: This is a follow up note on Gjergji Zaimi's fine response.
It seems that
$$\sum_nA_nq^n=\sum_k\frac{q^{2k-1}}{1-q^{2k-1}}=\sum_n\frac{d(n)}{\nu_2(2n)}q^n,$$
where $d(n)$ is the sum of (positive) divisors of $n$ and $\nu_2(m)$ is the $2$-adic valuation of $m$.
Therefore, we obtain the convolution sum
$$f_n=\sum_{k=1}^n\frac{d(k)\cdot p(n-k)}{\nu_2(2k)}$$
where $p(m)$ is the number of integer partition of $m$.