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{{inline|date=September 2021}}
In [[Mathematics|mathematics]], the '''Fibonomial coefficients''' or '''Fibonacci-binomial coefficients''' are defined as

:<math>\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}</math>

where ''n'' and ''k'' are non-negative integers, 0&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''n'', ''F<sub>j</sub>'' is the ''j''-th [[Fibonacci number]] and ''n''!<sub>F</sub> is the ''n''th [[Fibonorial]], i.e.

: <math>{n!}_F := \prod_{i=1}^n F_i,</math>

where 0!<sub>F</sub>, being the [[Empty product|empty product]], evaluates to&nbsp;1.

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

:<math>\binom{n}{0}_F = \binom{n}{n}_F = 1</math>

:<math>\binom{n}{1}_F = \binom{n}{n-1}_F = F_n</math>

:<math>\binom{n}{2}_F = \binom{n}{n-2}_F = \frac{F_n F_{n-1}}{F_2 F_1} = F_n F_{n-1},</math>

:<math>\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,</math>

:<math>\binom{n}{k}_F = \binom{n}{n-k}_F.</math>

==Fibonomial triangle==
The Fibonomial coefficients {{OEIS|id=A010048}} are similar to [[Binomial coefficient|binomial coefficient]]s and can be displayed in a triangle similar to [[Pascal's triangle]]. The first eight rows are shown below.

{| style='margin: 1em auto; text-align:center'
! width='40pt' | 
! width='30pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
! width='20pt' | 
|-
| <math>n=0</math> || colspan='8' | || colspan='2' | 1 || colspan ='7' | 
|-
| <math>n=1</math> || colspan='7' | || colspan='2' | 1 || colspan ='2' | 1 || colspan ='6' | 
|-
| <math>n=2</math> || colspan='6' | || colspan='2' | 1 || colspan ='2' | 1 || colspan ='2' | 1 || colspan ='5' | 
|-
| <math>n=3</math> || colspan='5' | || colspan='2' | 1 || colspan ='2' | 2 || colspan ='2' | 2 || colspan ='2' | 1 || colspan ='4' | 
|-
| <math>n=4</math> || colspan='4' | || colspan='2' | 1 || colspan ='2' | 3 || colspan ='2' | 6 || colspan ='2' | 3 || colspan ='2' | 1 || colspan ='3' | 
|-
| <math>n=5</math> || colspan='3' | || colspan='2' | 1 || colspan ='2' | 5 || colspan ='2' | 15 || colspan ='2' | 15 || colspan ='2' | 5 || colspan ='2' | 1 || colspan='2' | 
|-
| <math>n=6</math> || colspan='2' | || colspan='2' | 1 || colspan ='2' | 8 || colspan ='2' | 40 || colspan ='2' | 60 || colspan ='2' | 40 || colspan ='2' | 8 || colspan ='2' | 1 || 
|-
| <math>n=7</math> || colspan='1' | || colspan='2' | 1 || colspan ='2' | 13 || colspan ='2' | 104 || colspan ='2' | 260 || colspan ='2' | 260 || colspan ='2' | 104 || colspan ='2' | 13 || colspan ='2' | 1 
|}

The recurrence relation

:<math>\binom{n}{k}_F = F_{n-k+1} \binom{n-1}{k-1}_F  + F_{k-1} \binom{n-1}{k}_F </math>

implies that the Fibonomial coefficients are always integers.

The fibonomial coefficients can be expressed in terms of the [[Gaussian binomial coefficient|Gaussian binomial coefficients]] and the [[Golden ratio|golden ratio]] <math>\varphi=\frac{1+\sqrt5}2</math>:
:<math>{\binom n k}_F = \varphi^{k\,(n-k)}{\binom n k}_{-1/\varphi^2}</math>

==Applications==

Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive [[Biography:Fibonacci|Fibonacci]] numbers, namely Jarden proved that given any generalized Fibonacci sequence <math>G_n</math>, that is, a sequence that satisfies <math>G_n = G_{n-1} + G_{n-2}</math> for every <math>n,</math> then

:<math>\sum_{j = 0}^{k+1}(-1)^{j(j+1)/2}\binom{k+1}{j}_F G_{n-j}^k = 0,</math>

for every integer <math>n</math>, and every nonnegative integer <math>k</math>.

==References==
{{reflist}}
* {{Citation| last1=Benjamin| first1=Arthur T.| last2=Plott| first2=Sean S.|  title=A combinatorial approach to Fibonomial coefficients| url=http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf| location=Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711| access-date=2009-04-04| archive-url=https://web.archive.org/web/20130215012020/http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf| archive-date=2013-02-15| url-status=dead}}
* Ewa Krot, [https://doi.org/10.2478%2FBF02475975 ''An introduction to finite fibonomial calculus''], Institute of Computer Science, Bia lystok University, Poland.
* {{MathWorld|title=Fibonomial Coefficient|urlname=FibonomialCoefficient}}
* Dov Jarden, ''Recurring Sequences'' (second edition 1966), pages 30–33.

[[Category:Fibonacci numbers]]
[[Category:Factorial and binomial topics]]
[[Category:Triangles of numbers]]

{{Sourceattribution|Fibonomial coefficient}}