The power series expansion of certain infinite products q r ∏ n = 1 ∞ ( 1 − q n ) a 1 ( 1 − q 2 n ) a 2 ⋯ ( 1 − q m n ) a m $q^{r}\prod_{n=1}^{\infty}(1-q^{n})^{a_{1}}(1-q^{2n})^{a_{2}}\cdots(1-q^{mn})^{a_{m}}$

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