Let $x=(x_1,\ldots,x_n)\in X$ be a (column) vector and consider the operator $f: X\to\mathbb{R}$ that computes the sum of all unique pairwise multiplications of $x$. For example, for $n=3$, $f$ is defined as
$$f(x) = x_1x_2 + x_2x_3 + x_1x_3$$
I know we can write this as $f(x) = x^\top Ax$ where $A$ is the (strictly) upper triangular matrix of ones, but does this operator have a name (and consequently some studied properties)?