Geodesic
The generating function is rational for the number of rooted forests in a circulant graph
Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 129-137.

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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. 
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U. P. Kamalov; A. B. Kutbaev; A. D. Mednykh. The generating function is rational for the number of rooted forests in a circulant graph. Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 129-137. http://geodesic.mathdoc.fr/item/MT_2023_26_2_a5/

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