Define two sequences $(w_{n,k})_{n,k \geq 0}$ and $(w_n(k))_{n,k \geq 0}$ by the formulas
$$w_{n,k} = n! \sum_{i=0}^{n-k} \frac{u_i}{i!} \quad \text{and} \quad \sum_{n=0}^{\infty} w_n(k) x^n = \left(1 - s_1 x - \cdots - s_r x^r\right)^k \sum_{n=0}^{\infty} w_{n,k}\, x^n.$$
Show the equality $w_n(k) = v_n(k)$ for all $n$ and $k$ such that $n \geq kr$.