# Fibonomial coefficient

In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as

$\binom{n}{k}_F = \frac{F_nF_{n-1}\cdots F_{n-k+1}}{F_kF_{k-1}\cdots F_1} = \frac{n!_F}{k!_F (n-k)!_F}$

where n and k are non-negative integers, 0 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fibonorial, where 0!F, being the empty product, evaluates to 1.

The Fibonomial coefficients are all integers. Some special values are:

$\binom{n}{0}_F = \binom{n}{n}_F = 1$
$\binom{n}{1}_F = \binom{n}{n-1}_F = F_n$
$\binom{n}{2}_F = \binom{n}{n-2}_F = \frac{F_n F_{n-1}}{F_2 F_1} = F_n F_{n-1},$
$\binom{n}{3}_F = \binom{n}{n-3}_F = \frac{F_n F_{n-1} F_{n-2}}{F_3 F_2 F_1} = F_n F_{n-1} F_{n-2} /2,$
$\binom{n}{k}_F = \binom{n}{n-k}_F.$

The Fibonomial coefficients (sequence A010048 in OEIS) are similar to binomial coefficients and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below.

$n=0$ 1
$n=1$ 1 1
$n=2$ 1 1 1
$n=3$ 1 2 2 1
$n=4$ 1 3 6 3 1
$n=5$ 1 5 15 15 5 1
$n=6$ 1 8 40 60 40 8 1
$n=7$ 1 13 104 260 260 104 13 1

The recurrence relation

$\binom{n}{k}_F = F_{n-k+1} \binom{n-1}{k-1}_F + F_{k-1} \binom{n-1}{k}_F$

implies that the Fibonomial coefficients are always integers.