A Method for Determining the Mod-2k Behaviour of Recursive Sequences, with Applications to Subgroup Counting

We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß–Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of 2. ∗Research supported by the Austrian Science Foundation FWF, grant Y464-N18. †Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. ‡Research supported by Lise Meitner Grant M1201-N13 of the Austrian Science Foundation FWF. the electronic journal of combinatorics 18(2) (2012), #P37 1