This question concerns various important classes of formal power series. For concreteness and convenience, let us work with power series $F(x) = \sum_{n\geq 0}c_n x^n \in \mathbb{C}[[x]]$, i.e., with complex coefficients. Then we have the following inclusions: $$ \textrm{rational power series} \subseteq \textrm{algebraic power series} \subseteq \textrm{differentially finite power series}$$ See for instance the classic paper Differentially Finite Power Series by Stanley, which defines all these classes and explains the inclusions. These classes are especially important when thought of as generating functions of discrete structures according to size, e.g., words in some formal language according to their length.
When I conceptualize these classes of power series, I think of the following table:
| Power series class | Example $F(x)$ | Recurrence satisfied by $c_n$ | G.f. for formal language class |
|---|---|---|---|
| rationalrational | $\sum_{n\geq 0} 2^nx^n$ | linear recurrence with constant coefficients | regular language |
| algebraicalgebraic | $\sum_{n\geq 0}\binom{2n}{n}x^n$ | ??? | unambiguous context-free grammar |
| $D$-finite$D$-finite | $\sum_{n\geq 0}\binom{2n}{n}^2x^n$ | linear recurrence with polynomial coefficients | ??? |
Question: Are there reasonable things to fill in the missing "???" squares in the table above?