# Size of complement of context-free language

Let $$L$$ be a context-free language, $$\bar L$$ be its complement and $$\bar L_n$$ be the length $$n$$ words in $$\bar L_n$$.

What is known about $$|\bar L_n|$$?

Note that it is known that $$|L_n|$$ is either polynomial (if $$L$$ is bounded$$~$$), or grows exponentially. I wonder if anything similar might be true about $$|\bar L_n|$$. Warning, $$L$$ is allowed to be ambiguous!

From the proof that determining if a CFL $${L}$$ = $$\Sigma^*$$ is undecidable, the set of strings $$ID_0\#ID_1^R\#ID_2\#ID_3^R\#\ldots\#ID_t$$ where $$ID_0,ID_1,\ldots,ID_t$$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $$|\overline{L}_n|$$ can basically be any computable function less than exponential.