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 $\Gamma$ 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, 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]. 
Christian Krattenthaler (Universit\"at Wien) and Thomas W. M\"uller   (Queen Mary University of London)
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