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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 thelucanomial 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 andSagan defined the generalized Narayana numbers and conjectured that thesenumbers are positive integers for $n \geq 1$. In this paper we define theFiboNarayana number $N_{n,k,F}$ and give a new recurrence relation for both theFiboNarayana numbers and the generalized Narayana numbers, proving theconjecture that these are positive integers for $n \geq 1$.

A Recursion for the FiboNarayana and the Generalized Narayana Numbers