Find the coefficients of $\prod_i\frac{x_i+y}{1-e^{-(x_i+y)}}$

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I would like to compute the coefficients (in $y$) of the following series $$\prod_{i=1}^n\frac{x_i+y}{1-e^{-(x_i+y)}}$$ Is there a way to extract them using residue calculation?