The generating function is rational for the number of rooted forests in a circulant graph

We consider the generating function $\Phi$ for the number $f_\Gamma(n)$ of rooted spanning forests in the circulant graph $\Gamma$, where $\Phi(x)=\sum_{n=1}^\infty f_\Gamma(n)x^n$ and either $\Gamma=C_n(s_1,s_2,\dots,s_k)$ or $\Gamma=C_{2n}(s_1,s_2,\dots,s_k,n)$. We show that $\Phi$ is a rational function with integer coefficients that satisfies the condition $\Phi(x)=-\Phi(1/x)$. We illustrate this result by a series of examples.