-1^n by -n choose k-1 equals -1^k by -k choose n-1

Theorem
Let $n, k \in \Z_{\ge 0}$.

Then:
 * $\left({-1}\right)^n \dbinom {-n} {k - 1} = \left({-1}\right)^k \dbinom {-k} {n - 1}$