I am reading Richard Stanley's "Topics in Algebraic Combinatorics" and just before the notes for Chapter 8, he was discussing generating functions for plane partitions and solid partitions. It is claimed there that:
It is easy to see that for any integer sequence $a_0 = 1, a_1, a_2, \dots$, there are unique integers $b_1, b_2, \dots$ for which $$\sum\limits_{n\ge 0} a_n x^n = \prod\limits_{i\ge 1}(1-x^i)^{-b_i}$$
Not sure if I am missing something obvious, but this is certainly not "easy to see" for me. Any help will be appreciated.
Expaning the factors on the right-hand side as power series shows that they affect the coefficients $a_n$ only for $n\ge i$. Thus the $b_i$ can be iteratively determined for $i=1,2,3,\ldots$ , with each $b_i$ chosen such $a_i$ comes out right, without messing up $a_n$ for $n\lt i$.