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?
Generating function of non-regular language
regular languages