**Proof. **
We prove (1) using a generating-function approach.
We begin by dividing both sides of (1) by (*j*-1)! and
then rewriting the resulting left-hand side as

which, upon re-indexing, becomes

Replacing the binomial coefficient of right-hand side of (1) with the equivalent , and multiplying both sides by (-1)

The right-hand side (2) is simply the coefficient of

Thus, the coefficient of

Since

But, noticing that (-1)

which is the left hand side of (2). Thus both sides of (2) give the coefficient of