Exercise 2, § 4

Let $a_1, a_2, \ldots, a_p$ be integers $> 0$; denote by $\alpha_n$ the number of finite sequences $(x_i)_{1 \leq i \leq p}$ of integers $\geq 0$ satisfying the equation

Show that the formal power series $\sum_{n=0}^\infty \alpha_n X^n$ (over $\mathbf{Q}$) is the expansion of the rational fraction

EXERCISES