Ji-Cai Liu, Xue-Ting Jiang, Yeong-Nan Yeh: On super Catalan numbers modulo $8$, 201-208

Abstract:

In this note, we completely determine the half of the super Catalan numbers $S(m,n)={2m\choose m}{2n\choose n}/{m+n\choose m}$ modulo $8$, which generalizes Eu et al's congruence regarding Catalan numbers $C_n$ modulo $8$. In particular, we show that $S(m,n)/2\not\equiv_8 7$ for all positive integers $m$ and $n$. We also derive the half of central binomial coefficients ${2n\choose n}$ modulo $8$ as a corollary.

Key Words: Catalan numbers, super Catalan numbers, central binomial coefficients.

2020 Mathematics Subject Classification: Primary 05A10; Secondary 11B75.

Download the paper in pdf format here.