I'm trying to prove that if we have the elementary symmetric polynomials that the following identity holds:(where $x = (x_1,..,x_n)$ is a vector of n variables) $\sum_{k=0}^n e_k(x)^2 = x_1\cdots x_n \sum_{j=0}^{\lfloor n/2 \rfloor} {2j \choose j} e_{n-2j}(x_1+1/x_1,\cdots x_n+1/x_n).$ I'm trying to prove this combinatorially, but I'm stuck so I'd love some hints or tips.
Update: I guess we can get some interpretation with strictly increasing functions, but that thought isn't fully developed yet.