Let $e_k(x_1, \dots, x_n)$ be the $k$th elementary symmetric polynomial. It is known that
$$ \sum_{k=0}^n e_k(x_1, \dots, x_n) t^k = \prod_{i=1}^n (1 + x_i t). $$
I've encountered the following function
$$ f(x_1, \dots, x_n, t) = \frac{1}{\sum_{k=0}^n e_k(x_1, \dots, x_n) t^{(k^2)}} $$
In particular, I'm wondering if there is a way to express it as a product of polynomials. The roots of $\sum_{k=0}^n e_k(x_1, \dots, x_n) t^{(k^2)}$ are not that simple (at least, the results from Mathematica didn't look very promising), so expressing this as a product of monomials is probably not very useful.
Are there any approaches to express $f(x_1, \dots, x_n, t)$ as a product of polynomials?