I just want to know if I solved this problem correctly, thanks!
Find and verify a general formula for $\sum\limits_{k=0}^n k^p$ involving Stirling numbers of the second kind.
So I expanded $$\sum_{k=0}^n k^p = 0^p + 1^p + \cdots + n^p\tag{1}$$
and the Stirling numbers of the second kind can be represented as: $$n^p = \sum_{k=0}^n S(p,k)[n]_{k}\tag{2}$$
After replacing each term in $(1)$ by $(2)$, I should get:
$$\sum_{k=0}^n k^p = \sum_{k=0}^n S(p,k)[0]_{k} + \sum_{k=0}^n S(p,k)[1]_{k} + \cdots + \sum_{k=0}^n S(p,k)[n]_{k}\;.$$
Is this correct? How else am I supposed to verify this formula?