For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series
$$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$
Observe the equality: for all $n \geq r$ and $k \geq 0$,
$$v_n(k+1) = v_n(k) - s_1 v_{n-1}(k) - \cdots - s_r v_{n-r}(k).$$