Coefficients of a polynomial representation of factorials

Question

I'm trying to figure out the coefficients of $${(k+d)!}/{(k-n)!}$$ when expressed as a polynomial. Any ideas?

Answer 1

I presume this is supposed to be a polynomial in $k$. $$\dfrac{(k+d)!}{(k-n)!} = \prod_{j=-n+1}^d (k+j)$$ The coefficient of $k^i$ is the sum of the products of all subsets of $\{-n+1, \ldots, d\}$ having cardinality $d+n-i$.