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 \}$ .