Free subgroup numbers modulo prime powers: The non-periodic case

In [J. Algebra 452 (2016) 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group Γ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the number of free subgroups of index λ in Γ is - essentially - congruent to a binomial coefficient times a rational function in λ modulo a power of a prime that divides a certain invariant of the group Γ, respectively to a binomial sum involving such numbers. These results, apart from their intrinsic interest, in particular allow for a much more efficient computation of congruences for free subgroup numbers in these cases compared to the direct recursive computation of these numbers implied by the generating function results in [J. London Math. Soc. (2) 44 (1991) 75-94].