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?