# Fibonomial coefficient

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

${\displaystyle {\binom {n}{k}}_{F}={\frac {F_{n}F_{n-1}\cdots F_{n-k+1}}{F_{k}F_{k-1}\cdots F_{1}}}={\frac {n!_{F}}{k!_{F}(n-k)!_{F}}}}$

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.

## Special values

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

${\displaystyle {\binom {n}{0}}_{F}={\binom {n}{n}}_{F}=1}$
${\displaystyle {\binom {n}{1}}_{F}={\binom {n}{n-1}}_{F}=F_{n}}$
${\displaystyle {\binom {n}{2}}_{F}={\binom {n}{n-2}}_{F}={\frac {F_{n}F_{n-1}}{F_{2}F_{1}}}=F_{n}F_{n-1},}$
${\displaystyle {\binom {n}{3}}_{F}={\binom {n}{n-3}}_{F}={\frac {F_{n}F_{n-1}F_{n-2}}{F_{3}F_{2}F_{1}}}=F_{n}F_{n-1}F_{n-2}/2,}$
${\displaystyle {\binom {n}{k}}_{F}={\binom {n}{n-k}}_{F}.}$

## Fibonomial triangle

The Fibonomial coefficients (sequence A010048 in the 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.

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

The recurrence relation

${\displaystyle {\binom {n}{k}}_{F}=F_{n-k+1}{\binom {n-1}{k-1}}_{F}+F_{k-1}{\binom {n-1}{k}}_{F}}$

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 ${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}$:

${\displaystyle {\binom {n}{k}}_{F}=\phi ^{k\,(n-k)}{\binom {n}{k}}_{-1/\phi ^{2}}}$