On the $3$-adic valuation of the Narayana numbers

We define the Narayana sequence $\{a_n{\}}_{n\ge 0}$ as the one satisfying the linear recurrence relation $a_n=a_{n-1}+a_{n-3}$ for $n\ge 3$, with initial values $a_0=0$ and $a_1=a_2=1$. In this paper, we fully characterize the $3$-adic valuation of $a_n$ and use this to determine all Narayana numbers that are factorials.