define the difference operator D A = {a[i+1]-a[i]}

The set of sequences where a[i] = some polynomial in i = P(i) is a vector space in X.
The difference operator D is linear: for scalars k and l, and polynomials p and q
D (kp+lq) = kDp+llDq
The the polynomials p(k,n) = n, n(n+1)/2, n(n+1)(n+2)/3, … n(n+1)(n+2)..(n+k-1)/k!
all satisfy the relation D p(k,n) = p(k-1,n) … easily proved by induction.
furthermore D is nilpotent on any particular polynomial. i.e. there exists k, D^k( p )= 0
we know D^k p(k,n) = 1
D^k (q) = 0 for polynomials of lower degree.

now p(k,n) = n^k/k! + lower order terms
D^k p(k,n) = 1 = (1/k!) * D^k (n^k)
D^k n^k = k!

Now go to the binomial theorem: Taking derivatives and plugging in gives you 0 on the left, and the terms of the inner sum on the right. At least for the first derivatives. The -th derivative gives you on the left, and the very last term of the inner sum (and only for one value of the outer sum). So everything cancels except for one term:

I’m really waiting to see how you’ll do it combinatorially.