  Given a finitely generated group with generating set $S$, we study the
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 the corresponding Cayley graph. Muller and Schupp proved the generating
function of the sequence is algebraic when $G$ has a finite-index free subgroup
(using a result of Dunwoody). In this work we make this result effective for
free products of finite groups: we determine bounds for the degree and height
of the minimal polynomial of the generating function, and determine the minimal
polynomial explicitly for some families of free products. Using these results
we are able to prove that a gap theorem holds: if $S$ is a finite symmetric
generating set for a group $G$ and if $a_n$ denotes the number of words of
length $n$ over the alphabet $S$ that are equal to $1$ then $\limsup_n
a_n^{1/n}$ exists and is either $1$, $2$, or at least $2\sqrt{2}$.

Combinatorics