 Euler transform of a sequence

The Euler transform of a sequence $(b_n{)}_{n\ge 1}$ has coefficients $a_n$ defined by ${\sum }_{n\ge 0}{a}_n{x}^n={\prod }_{n\ge 1}(1-x^n{)}^{-b_n}$. In unlabelled counting, $b_n$ may count connected components of size $n$, while $a_n$ counts multisets of components.