Structure and asymptotics for Catalan numbers modulo primes using automata

Abstract

Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n:C_n\equiv 0\mathrm{mod}p\}$ is $1$ for all primes $p$, We also show that if $n=p^k-1$ then $C_n\equiv -1\mathrm{mod}p$. Finally we show that if $n\equiv \{\frac{p+1}{2},\frac{p+3}{2},...,p-2\}\mathrm{mod}p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.

15 pages, 2 tables, 3 figures

References in corpus (3)

• Asymptotic density of Motzkin numbers modulo small primes
• Structure and asymptotics for Motzkin numbers modulo small primes using automata
• Asymptotic density of Catalan numbers modulo 3 and powers of 2

Cited by in corpus (1)

• The Catalan numbers have no forbidden residue modulo primes