Macdonald's formula for symmetric products

Statement

Let X be a finite CW complex, of dimension d, with Poincaré polynomial

$$P_X (t) = \sum_{k=0}^d b_k(X) \, t^k$$

where $b_k(X)$ is its kth Betti number. Then the Poincaré polynomial of the nth symmetric product of X, denoted $\operatorname{Sym}^n (X)$, is obtained from the series expansion:

$$\operatorname{PE}[P_X(-t)\,x] = \prod_{k=0}^d (1-t^k x)^{(-1)^{k+1}b_k(X)} = \sum_{n\geq 0} P_{\operatorname{Sym}^n(X)}(-t) \, x^n$$