  Given a finitely generated group with generating set $S$, we study the
\emph{cogrowth} sequence, which is the number of words of length $n$ over the
alphabet $S$ that are equal to one. This is related to the probability of
return for walks in a Cayley graph with steps from $S$. We prove that the
cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of
superpolynomial growth, answering a question of Garrabant and Pak.
