For language $L$ over an alphabet $\Sigma$ denote $\gamma_L(n)$ as the number of words of length $n$ in the language $L$. It is known that for regular languages this function represents a sequence with rational generating function (which is equivalent to that $\gamma_L(n)$ is linear-recurrent for sufficiently large coefficients).
However, I couldn't find any information about non-regular languages. And it is not clear how to extend the result, stated above to some other types of languages.
Does anyone know the conditions for some non-regular classes of languages (for example, prefix-closed languages) to have rational geodesic growth function?