关于FiboNarayana和广义Narayana数的一个递推式
A Recursion for the FiboNarayana and the Generalized Narayana Numbers
摘要：
卢卡斯多项式$\{n\}$是由$\{ n \} = s \{ n-1 \} + t \{ n-2 \}$针对具有$ \{ 0 \} = 0$和$\{ 1 \} = 1$的$n \geq 2$给出的$s$和$t$中的多项式。Lucanomial系数是二项式系数的模拟，由\[ \Bigl\{ \begin{array}{c} n\\k \end{array} \Bigr \} = \frac{ \{n\}! }{ \{k\}! \{n-k\}!}. \]给出当$s = t = 1$然后$\{ n \} = F_n$和Lucanomial系数变成多项式系数\[ \binom{n}{k}_F = \frac{F_n!}{F_k! F_{n-k}!}. \]著名的纳拉亚纳数$N_{n,k}$满足方程\[ N_{n,k} = \frac{1}{n} \binom{n}{k} \binom{n}{k-1}. \] \[ %C_n = \sum_{k=1}^n N_{n,k}. %\]在2018年，Bennett，Carrillo，Machacek和Sagan定义了广义纳拉亚纳数，并证明这些数是$n \geq 1$的正整数。本文定义了FiboNarayana数$N_{n,k,F}$，给出了FiboNarayana数和广义Narayana数的一个新的递推关系，证明了$n \geq 1$的正整数猜想.
英文摘要：
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 $n \geq 1$.
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