Exercise 3, § 4

Let F be a finite set of integers > 0, and let $\beta_n$ be the number of finite sequences $(x_i)$ of at most $n$ terms, all of whose terms lie in F and such that $\sum x_i = n$. Show that the formal power series $\sum_{n=0}^m \beta_n X^n$ over $Q$ is the expansion of the rational fraction