# Number of words of length n for special language

Let $$\Sigma$$ be an alphabet and let $$L$$ be a language over it with the following properties:

1. if $$w\in L$$ then there exists $$v\in \Sigma^*$$ such that $$wv \in L$$ and for every $$s\in \Sigma$$ the word $$wvs$$ does not lie in $$L$$
2. $$wv\in L$$ then $$vw \in L$$
3. It is prefix-closed, i.e. prefix of any word is still in the language.

Note that by the definition, it is not cyclic language. I’m trying to compute its growth function, by that I mean $$\gamma_n:= |\{w\in L \mid |w| = n\}|$$. I know about my specific case that it is not regular and my hypothesis is that function $$\Gamma(x) = \sum_{n=1}^\infty \gamma_nx^n$$ is not rational. However, I couldn’t find any information about these functions for non-regular languages. Maybe, there’s a formula that connects entropy of language, i.e. $$e(L):= \limsup\limits_{n\to\infty} \frac{\log\gamma_n}{n}$$ and the $$\Gamma$$ function. Or for such a language there’s a way to describe its growth throughout the growth of the language $$\operatorname{End}(L) = \{ w\in L \mid \forall s\in \Sigma \,ws \text{ is not in } L \}$$.