Evaluation of the symmetrized Mordell-Tornheim zeta function

In this paper we evaluate the symmetrized Mordell-Tornheim zeta function defined as \begin{equation*} \overline{\zeta}_n(w_1, \ldots, w_n) = \sum_{\substack{a_1, \ldots, a_n \in \mathbb{Z}^* \\ a_1 + \ldots + a_n = 0}} \frac{1}{\left| a_1^{w_1} \cdots a_n^{w_n} \right|} \end{equation*} where $n \ge 1$ is a positive integer representing the depth and $w_1, \ldots, w_n \ge 1$ are positive integers representing the weight $w = w_1 + \ldots + w_n$ of the function. Compared to the classical Mordell-Tornheim zeta function $\zeta_{MT,n}(w_1, \ldots, w_n; w_{n+1})$ which is restricted to the positive orthant (hyperoctant), the symmetrized one spans the entire $(n-1)$-dimensional hyperplane. We show that when the depth and the weight of the function are equal, that is for $\overline{\zeta}_n(1, \ldots, 1)$, it has a remarkably simple representation in terms of standard functions: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(f^{(1)}(0), \ldots, f^{(n)}(0)) \end{equation*} where $B_n$ is $n$-th complete exponential Bell polynomial and $f^{(n)}(0)$ is $n$-th derivative at $x=0$ of function $f(x)$ defined as: \begin{equation*} f(x) = \ln \binom{-2x}{-x} \end{equation*} Additionally, we show the value can be expressed using the following polynomials with positive integer coefficients over the values of zeta function: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(0, (2^2 - 2) \Gamma(2) \zeta(2), \ldots, (2^n - 2) \Gamma(n) \zeta(n)) \end{equation*} or equivalently, over the values of eta function: \begin{equation*} \overline{\zeta}_n(1, \ldots, 1) = B_n(0, 2^2 \Gamma(2) \eta(2), \ldots, 2^n \Gamma(n) \eta(n)) \end{equation*} The list of explicit values for small $1 \le n \le 10$ is available in the appendix.