# reference request – Alternating sum of hook lengths

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