The Hilbert-Poincare series for some algebras of invariants of cyclic groups

Let $\rho$ be a linear representation of a finite group over a field of characteristic 0. Further, let $R_{\rho}$ be the corresponding algebra of invariants and let $P_{\rho}(t)$ be its Hilbert-Poincare series. Then the series $P_{\rho}(t)$ presents a rational function $\Psi(t)/\Theta(t)$. If $R_{\rho}$ is a complete intersection then $\Psi(t)$ is a product of cyclotomic polynomials. Here we prove the inverse statement for the case when $\rho$ is “almost regular” (in particular, regular) representation of a cyclic group. It gives the answer to a question of R. Stanley in this very particular case.