Denote $\pmb{X}_n=(x_1,x_2,\dots,x_n)$. Consider the symmetric polynomial $$f_n(\pmb X_n)=\prod_{1\leq i<j\leq n}(x_i+x_j).$$ Expand these in terms of elementary symmetric polynomials, say $$f_n(\pmb{X}_n)=\sum_{\mu}c_{\mu,n}\cdot e_{\mu}(\pmb{X}_n).$$

For example, \begin{align*} f_3&=-e_{(3)}+e_{(2,1)} \\ f_5&=-e_{(5,5)}+2e_{(5,4,1)}+e_{(5,3,2)}-e_{(5,2,2,1)}-e_{(4,4,1,1)}-e_{(4,3,3)}+e_{(4,3,2,1)}. \end{align*}

QUESTION 1.Is it true that, for integers $n \geq 1$, we have $$\sum_{\mu}c_{\mu,2n+1}=0?$$

**POSTSCRIPT.** Fedor's reply (to Question 1) shown below suggests to me to ask:

QUESTION 2.Is it true that, for integers $n \geq 1$, we have $$\sum_{\mu}c_{\mu,2n}=(-1)^{\binom{n}2}?$$