Geodesic
Cogrowth series of free products of finite and free groups
Glasgow mathematical journal, Tome 41 (1999) no. 1, pp. 19-31

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Let A={a1,...,an,a1−1,...,an−1} and iteration of A denoted by A[starf ] to be the set of words in A (including the empty word). Let S⊆A[starf ]; then the growth function of the set S is the function Γ(l)=number of words in S of length l. For m≤n let $\vec {a}$=(ai1,...,aim), where ik∈{1,...,n} are different; then the relative growth function with respect to $\vec {a}$ is the function Γ$\vec {a}$(l,l1,...,lm)=number of words in S of length l+l1+...+lm having (for each k) lk total occurrences of aik and aik−1. 
KUKSOV, DMITRI. Cogrowth series of free products of finite and free groups. Glasgow mathematical journal, Tome 41 (1999) no. 1, pp. 19-31. doi: 10.1017/S001708959997026X
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