The sum:
$$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$
Can be calculated by this formula, called the "Bernoulli formula" in wikipedia
$$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+1-k} $$
I tried to find an intuitive proof of this formula (not by induction or anything) but I didn't find any. Does somebody knows any proof or can help me with some link?
Thank you so much!