A method for determining the mod-$p^k$ behaviour of recursive sequences

Abstract

We present a method for obtaining congruences modulo powers of a prime number~$p$ for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors [Electron. J. Combin. 8(2) (2012), Art. P37; arXiv:1107.2015] from $p=2$ to arbitrary primes. Our applications include congruences for numbers of non-crossing graphs and numbers of Kreweras walks modulo powers of~$3$, as well as congruences for Fuß-Catalan numbers and blossom tree numbers modulo powers of arbitrary primes.

AmS-LaTeX, 36 pages. Several typos corrected. arXiv admin note: substantial text overlap with arXiv:1107.2015; text overlap with arXiv:1308.2856