TITLE: Expansion of $n(n-1)(n-2)...(n-k)$
QUESTION [5 upvotes]: Is there a way to express the following such that each coefficient of the expansion can be found selectively:
$$n(n-1)(n-2)...(n-k)$$
For example, the first term is obviously $n^{k+1}$ and so it's coefficient is $1$ and the last term is $(-1)^k \cdot k! \cdot n$ which has coefficient $(-1)^k \cdot k!$.
I am attempting to generalise the nth derivative of a probability generating function evaluated at $1$ in terms of $E(X^n)$ and this expansion will allow the result to be generalised.

REPLY [2 votes]: Yes there is a way, i think vieta's formulae is what you are looking for. 
https://en.m.wikipedia.org/wiki/Vieta%27s_formulas