Number of words of length n in a context-free language Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language.


  What is known about $w_n$?


I'm sure this has been studied a lot, but I couldn't find anything at all on it.
Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser:

Either there is a polynomial $p$ so that $w_n\le p(n)$ for all $n$
Or there exists a $c&gt;1$, so that $w_n\ge c^n$ for infinitely many $n$.

This has been shown for instance in:

Roberto Incitti:
&quot;The growth function of context-free languages&quot;
Theoretical Computer Science 255 (2001), Pages 601-605
Martin R. Bridson, Robert H. Gilman:
&quot;Context-Free Languages of Sub-exponential Growth&quot;
Journal of Computer and System Sciences 64 (2002), Pages 308-310

And for a given context-free grammar, one can decide in polynomial time whether the generated language has polynomial or exponential growth:

Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:
&quot;Finding the Growth Rate of a Regular or Context-Free Language in Polynomial Time.
International Journal of Foundations of Computer Science 21 (2010), Pages 597-618