I arrived at this formula by inductive reasoning, but I don’t know how to prove it.

For any natural numbers $m$ and $k=0,1,2,\ldots, m-1$, $B_i$ - Bernoulli numbers we have: $$\sum_{i=0}^k (-1)^{k-i}\cdot B_i\cdot\binom{m+i-k-1}{i}\cdot\binom{m}{m+i-k-1}=(-1)^{k}\cdot m\cdot\binom{m-1}{k}$$