General formula for the sum of $k$-products

Question

I try to obtain a formula for the sum $$\sum_{1\le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \dots i_k$$ with $1\le k \le n$.

The case $k=2$ is asked here. Following the hint there, I expand $(1+2+\dots+n)^2$ and using the formula for $\sum_{i=1}^{n} i^2$ I was able to get the formula. For the case $k=3$, I expanded $(1+2+\dots+n)^3$ and by adding some terms I obtained a formula -which took a lot more time than the latter case-. However, I wonder how can I obtain a general formula?

Answer 1

By Vieta's formulas, the sum equals the coefficient of $x^{n-k}$ in $(x+1)\cdots(x+n)$, which is equal to ${n+1\brack n+1-k}$ using Stirling numbers of the first kind (compare with this result).