Let $lambda=(lambda_1geqlambda_2geqcdotsgeqlambda_k>0)$ be an integer partition and $lambda’$ be its conjugate partition. Denote the hook length of a cell $(i,j)$ in the Young diagram of $lambda$ by $h_{ij}(lambda)$.

QUESTION.Is there a formula for the following sequence? Or, a generating function?

$$f_n=sum_{lambdavdash n}sum_{(i,j)inlambda}(-1)^{i+j}h_{ij}(lambda).$$