Let $n$ be a positive integer. For each $k = 1, \ldots, n$, let $$ S_k(n) := \sum_{1 \le i_1 < \cdots < i_k \le n} i_1 \cdots i_k $$ be the sum of the $k$-wise product of distinct integers from $1$ to $n$. Does this sum $S_k(n)$ have a name and formula?

For example, $$S_1(n) = 1 + \cdots + n = \binom{n + 1}{2}$$ and $$S_2(n) = [(1 + \cdots + n)^2 - (1^2 + \cdots + n^2)]/2 = n(n+1)(3n^2 - n - 2)/24.$$