Let $\lambda$ be a partition, represented by a usual Young diagram in which $1\le i\le \ell(\lambda)$ labels the rows and, for each $i$, $1\le j\le \lambda_i$ labels the columns. For each box $\square$ in the diagram, $c(\square)=j-i$ is its content. The polynomial $$ P_\lambda(x)=\prod_{\square\in\lambda}(x+c(\square))=\prod_{(i,j)\in\lambda}(x+j-i)$$ is the content polynomial.

I would like to know some family of coefficients $b^\lambda_{\mu\nu}$, as a function of three partitions, such that $$P_\lambda(x+y)=\sum_{\mu,\nu\subset \lambda}b^\lambda_{\mu\nu}P_\mu(x)P_\nu(y).$$

(Notice that content polynomials are not linearly independent, so this equation alone is not enough to uniquely determine the coefficients. I ask for *some* family)