Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in:

$$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$

or is it hopeless?

I can't think of any other method than trying to expand each factor.

Background: the above polynomial is the generating function for a system of linear equations in binary values (see this question).

Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in:

$$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$

or is it hopeless?

I can't think of any other method than trying to expand each factor.

Background: the above polynomial is the generating function for a system of linear equations in binary values (see this question).

For the simplest case of the coefficient of $z_1^kz_2^k$ in $[(1+z_1)(1+z_2)(1+z_1z_2)]^k$ I found the formula $\sum_{j=0}^k \binom{k}{j}^3$ at OEIS A000172.