Let $L$ be a language on a finite alphabet and let $L\_n$ be the number of words of length $n$. Let $f\_L(x) = \sum\_{n \ge 0} L\_n x^n$. The following are well-known:


* If $L$ is regular, then $f\_L$ is rational.
* If $L$ is unambiguous and context-free, then $f\_L$ is algebraic.


Does there exist a natural family of languages $\mathcal{L}$ containing the context-free languages such that if $L \in \mathcal{L}$, then $f\_L$ is [holonomic](http://en.wikipedia.org/wiki/Holonomic#Holonomic_function)? Is that class of languages also associated to a natural class of automata? 


This question is prompted by a remark in Flajolet and Sedgewick where they assert that there is no meaningful generating function formalism associated to context-sensitive languages because of the significant undecidability issues. However, holonomic functions have proven a robust and incredibly useful framework in combinatorics, so I think this is a natural question to ask.



Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?
I vaguely remember that the paper by Marni Mishna and Mike Zabrocki [Analytic aspects of the shuffle product](http://arxiv.org/abs/0802.2844) sheds some light on the subject.


