Divided differences of polynomials

Question

I am interested in computing divided differences of polynomials in a numerically stable way. Therefore I want to prove/disprove the following formula: $$\left(x^{n+m}\right)[x_1,\dots,x_n] = \sum_{1\leq i_0\leq\dots\leq i_m\leq n} x_{i_0}\dots x_{i_m} \qquad \forall x_1,\dots, x_n\in\mathbb{R} \text{ and } \forall n\in\mathbb{N} \text{ and } \forall m\in\mathbb{N}_0.$$ I have tried to prove it by induction on $n$. I have shown it is true for a few small values of $m$. Does someone have a clue whether the equation holds in general?