Integral zeros of a polynomial with linear recurrences as coefficients

Let $K$ be a number field, $S$ a finite set of places of $K$, and $\mathcal{O}_S$ be the ring of $S$-integers. Moreover, let $$G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)}$$ be a polynomial in $Z$ having simple linear recurrences of integers evaluated at $n$ as coefficients. Assuming some technical conditions we give a description of the zeros $(n,z) \in \mathbb{N} \times \mathcal{O}_S$ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.