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, i.e.


 * $${n!}_F := \prod_{i=1}^n F_i,$$

where 0!F, being the empty product, evaluates to 1.

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


 * $$\binom{n}{0}_F = \binom{n}{n}_F = 1$$


 * $$\binom{n}{1}_F = \binom{n}{n-1}_F = F_n$$


 * $$\binom{n}{2}_F = \binom{n}{n-2}_F = \frac{F_n F_{n-1}}{F_2 F_1} = F_n F_{n-1},$$


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


 * $$\binom{n}{k}_F = \binom{n}{n-k}_F.$$

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

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.

The fibonomial coefficients can be expressed in terms of the Gaussian binomial coefficients and the golden ratio $$\varphi=\frac{1+\sqrt5}2$$:
 * $${\binom n k}_F = \varphi^{k\,(n-k)}{\binom n k}_{-1/\varphi^2}$$

Applications
Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence $$G_n$$, that is, a sequence that satisfies $$G_n = G_{n-1} + G_{n-2}$$ for every $$n,$$ then


 * $$\sum_{j = 0}^{k+1}(-1)^{j(j+1)/2}\binom{k+1}{j}_F G_{n-j}^k = 0,$$

for every integer $$n$$, and every nonnegative integer $$k$$.