A Short Approach to Catalan Numbers Modulo 2r

We notice that two combinatorial interpretations of the well-known Catalan numbers Cn = (2n)!/n!(n+1)! naturally give rise to a recursion for Cn. This recursion is ideal for the study of the congruences of Cn modulo 2 r, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh’s recent classification of Cn modulo 2 r. The equivalence Cn ≡2r Cn̄ is further reduced to Cn ≡2r Cñ for simpler ñ. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose 2-adic order is equal to that of Cn, which is one less than the sum of the digits of the binary expansion of n + 1.