Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers

Theorem
Let $n, k \in \Z_{\ge 0}$ be positive integers.

Let $S_k = \displaystyle \sum_{i \mathop = 1}^n i^k$.

Then:
 * $\displaystyle \sum_{i \mathop = 1}^k \binom {k + 1} i S_i = \left({n + 1}\right)^{k + 1} - \left({n + 1}\right)$