The number of terms of the polynomial:
$$\prod_{1\leq i<j\leq n}\left( x_{i}+x_{j}\right)$$
coincides for $n=2,3,4,5,6$ with the sequence OEIS A001858. One of the things it counts is the number of forests of trees on $n$ labeled nodes.
I tried to check some of the references at the OEIS page but up to now I haven't found any reference to the above polynomial.
I have found also some ways of expressing the polynomial in this answer but I am not sure it can help here.
Any hint for proving that the OEIS sequence counts also the number of terms of the above polynomial for any $n \ge 2$?