On a conjecture of J. Shallit about Apéry-like numbers

Abstract

Put $a(n)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$, and $b(n)=v_3(a(n))$, for all integers $n⩾0$, 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 $(b(n){)}_{n⩾0}$ is $3$-regular.

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