# Growth function for non-regular languages

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?