On a conjecture of J. Shallit about Ap\'ery-like numbers

Abstract

Put $a(n)=\sum\limits_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}$, and $b(n)=v_{3}(a(n))$, for all integers $n\geqslant 0$, where $v_{3}$ is the $3$-adic valuation. In this work, we shall confirm a formula about $b(n)$, conjectured by J. Shallit in 2000. As application, we show that the sequence $(b(n))_{n\geqslant 0}$ is $3$-regular.