Generating function of non-regular language Let's say $\sum_{n \ge 1} a_{n}x^n$ is generating function for regular language $L$. $a_{n}$ is number of words with length $n$. 

Find an example of generating function which isn't correspond for any regular language.

My attempt : I use the fact for any regular language there is exists $n_{0}$ $\lambda_{i}$ and $p_{i}$ : $a_{n} = \sum_{i} \lambda_{i}^{n} p_{i}$ and as example get language with $a_{n} = C_{n}$ , where $C_{n}$ is Catalan number. My teacher said that's not obvious and told me to get easier example. But I don't know useful criteria to find an example with contradiction about regularity. Any ideas?
If we apply a substitution that maps all letters of the alphabet to the same letter, we obtain a unary language. This shows that if $L$ is regular, then the set $\{n : L \cap \Sigma^n \neq \emptyset\}$ is eventually periodic. This shows that $\sum_{n=0}^\infty x^{n^2}$ corresponds to no regular language, for example.