A method for determining the mod-$2^k$ behaviour of recursive sequences, with applications to subgroup counting

TL;DR

The paper introduces a method to analyze the mod-$2^k$ behavior of recursive sequences with polynomial coefficients, applying it to classical combinatorial sequences and subgroup counting functions, yielding new congruence results.

Contribution

It provides a novel technique for deriving congruences modulo powers of 2 for a broad class of recursive sequences, extending known results to higher powers.

Findings

New congruences for Catalan numbers modulo 2^k

Extensions of subgroup counting results to higher powers of 2

Application of the method to Fu extss-Catalan numbers and Hecke groups

Abstract

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, Fu\ss-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.