TITLE: Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?
QUESTION [17 upvotes]: 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?  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.

REPLY [2 votes]: There are such natural examples, such as the very recent ICALP'20 contribution on "unambiguous finite and Pushdown Parikh automata" (paper not yet available),
or the related RCM class by Massazza and Castiglione.
In both cases the generating function of these language classes are holonomic and moreover contain examples which are not algebraic.
EDIT: The ICALP'20 paper is now available.