A Recursion for the FiboNarayana and the Generalized Narayana Numbers

Abstract: The Lucas polynomials, ${n}$, are polynomials in $s$ and $t$ given by ${ n } = s { n-1 } + t { n-2 }$ for $n \geq 2$ with ${ 0 } = 0$ and ${ 1 } = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are given by [ \Bigl{ \begin{array}{c} n\k \end{array} \Bigr } = \frac{ {n}! }{ {k}! {n-k}!}. ] When $s = t = 1$ then ${ n } = F_n$ and the lucanomial coefficient becomes the fibonomial coefficient [ \binom{n}{k}F = \frac{F_n!}{F_k! F{n-k}!}. ] The well-known Narayana numbers, $N_{n,k}$ satisfy the equation [ N_{n,k} = \frac{1}{n} \binom{n}{k} \binom{n}{k-1}. ] [ %C_n = \sum_{k=1}^n N_{n,k}. %] In 2018, Bennett, Carrillo, Machacek and Sagan defined the generalized Narayana numbers and conjectured that these numbers are positive integers for $n \geq 1$. In this paper we define the FiboNarayana number $N_{n,k,F}$ and give a new recurrence relation for both the FiboNarayana numbers and the generalized Narayana numbers, proving the conjecture that these are positive integers for $n \geq 1$.