Let f(x) be a polynomial with order m.
Denote $\Delta f(x) = f(x + 1) - f(x)$ the difference, and $\Delta^k f(x) = \Delta^{k-1} f(x+1) - \Delta^{k-1} f(x)$.
Then we have the formula $$\begin{aligned} \sum_{k=1}^n f(k) &= \binom{n}{1} f(1)+\binom{n}{2}\Delta f(1) + \binom{n}{3} \Delta^3 f(1)+\dots+ \binom{n}{m+1} \Delta^mf(1)\\ &=\sum_{k=1}^{m+1} \binom{n}{k} \Delta^{k-1} f(1) \end{aligned}$$
Does this formula has a name? How to proof it? Any help is appreciated!