# Free subgroup numbers modulo prime powers: the non-periodic case

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Abstract. In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group \Gamma 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 \lambda in \Gamma is - essentially - congruent to a binomial coefficient times a rational function in \lambda modulo a power of a prime that divides a certain invariant of the group \Gamma, respectively to a binomial sum involving such numbers. These results 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].

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