# Is the spherical growth function of a finite group always a unimodular sequence?

Let $$G$$ be a finite group with symmetric generating set $$S$$, and let the spherical growth function of $$G$$ (with respect to those generators) be $$a_S(n) = |\{g \in G \mid \ell_S(g) = n \}|$$, where $$\ell_s(g)$$ is word length of $$g$$ with respect to $$S$$.

For a typical choice of generators (say, transposistions in $$S_n$$) one expects the sequence $$a_S(n)$$ to increase early on and then later decrease. But is it possible for the sequence to exhibit more complicated behavior, such as increasing, then decreasing, then increasing again (that is, for the sequence to not be unimodular)?

It is known that there are infinite finitely generated groups without monotonic growth function, and this seems a natural question to ask in the finite case.

I just calculated the growth series for $$A_5$$ with generators $$(1,2)(3,4)$$ and $$(2,3,5)$$, and it is $$1,\,3,\,4,\,6,\,8,\,10,\,8,\,10,\,6,\,3,\,1,$$ so it has a little dip in the middle.