I've worked out a few summation formulas, and I am hoping to find a pattern. Unless I have made a mistake somewhere, we have the following identities:
$$\sum_{1 \le i \le n} i = \frac{(n+1)n}{2} $$
$$\sum_{1 \le i < j \le n} ij = \frac{(3n+2)(n+1) \, n \, ( n-1)}{24}$$
$$\sum_{1 \le i < j < k \le n} ijk = \frac{(n+1)^2 \, n^2 \, (n-1)(n-2) }{48}$$
It seems clear that $$p_k(n)= \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $$ is a polynomial in $n$ of degree $2k$. But is there a nice closed-form formula for it? Maybe in terms of its factorization?
Comment: I realize that what I am looking for is the coefficient of $t^k$ in the expansion $$(1+t)(1+2t) \dots (1+nt),$$ so maybe generating function techniques could be helpful. But the main way I know to extract the coefficient of $t^k$ is to take derivatives $k-1$ times and on the surface of things that looks like a huge mess.