Coefficients of the expansion of $\prod_{i=1}^k(x+i)$

Question

This seems to be something well known, but I couldn't find any reference.

Suppose that we wish to expand the product $\prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+\ldots+a_{k-1}x+ a_k$. The coefficients of this expansion would be: $$\begin{aligned} a_0={ }&1\\ a_1={ }&\sum_{i=1}^k i=\frac{k(k+1)}{2}\\ a_2={ }&\frac{1}{2}\sum_{1\le i,j\le k\atop i\neq j}ij=\frac{1}{2}\left[\left(\frac{k(k+1)}{2}\right)^2-\sum_{i=1}^k i^2\right]\\ \ldots\\ a_k={ }&\prod_{i=1}^k i=k!\end{aligned}$$ I wonder if there is a name and an expression for the coefficients $a_i$?

Answer 1

One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,\dots, k)$,in $1,\dots, k$.