Consider the expansion of the following $N$ variable expression $$ D_N(z_1,\ldots,z_N)=\prod_{1\leq j<k\leq N}\left(1-\frac{z_j}{z_k}\right)\left( 1-\frac{z_k}{z_j} \right) $$ For example, in the case of $N=3$ we have $$ \begin{align*} D_3(z_1,z_2,z_3) & = \left(1-\frac{z_1}{z_2}\right)\left( 1-\frac{z_2}{z_1} \right) \left(1-\frac{z_2}{z_3}\right)\left( 1-\frac{z_3}{z_2} \right) \left(1-\frac{z_3}{z_1}\right)\left( 1-\frac{z_1}{z_3} \right) = \\ & = 6 -2\frac{z_1}{z_2}-2\frac{z_2}{z_1} -2\frac{z_2}{z_3}-2\frac{z_3}{z_2} -2\frac{z_3}{z_1}-2\frac{z_1}{z_3} + \\ & +2\frac{z_1z_2}{z_3^2} +2\frac{z_3^2}{z_1z_2} +2\frac{z_2z_3}{z_1^2} +2\frac{z_1^2}{z_2z_3} +2\frac{z_3z_1}{z_2^2} +2\frac{z_2^2}{z_3z_1} - \\ & -\frac{z_1^2}{z_2^2} -\frac{z_2^2}{z_1^2} -\frac{z_2^2}{z_3^2} -\frac{z_3^2}{z_2^2} -\frac{z_3^2}{z_1^2} -\frac{z_1^2}{z_3^2} \end{align*} $$

It is known that the constant term of $D_N$ is $N!$ (this can be proved using the formula for the Dyson integral). I am interested in finding out if there are any patterns that describe the coefficients of the other (non-constant) terms in the expansion of $D_N$ ?

As an analogy for what I'm hoping to obtain, in the case of the well-kown multinomial expansion $$ (z_1+z_2+\ldots+z_M)^N $$ the coefficient of any term $z_1^{k_1}z_2^{k_2}\ldots z_M^{k_M}$ can be simply expressed as $$ \frac{N!}{k_1! k_2!\ldots k_M!} $$