Fibonomial coefficient
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In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as
- [math]\displaystyle{ \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 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fibonorial, i.e.
- [math]\displaystyle{ {n!}_F := \prod_{i=1}^n F_i, }[/math]
where 0!F, being the empty product, evaluates to 1.
Special values
The Fibonomial coefficients are all integers. Some special values are:
- [math]\displaystyle{ \binom{n}{0}_F = \binom{n}{n}_F = 1 }[/math]
- [math]\displaystyle{ \binom{n}{1}_F = \binom{n}{n-1}_F = F_n }[/math]
- [math]\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}, }[/math]
- [math]\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, }[/math]
- [math]\displaystyle{ \binom{n}{k}_F = \binom{n}{n-k}_F. }[/math]
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.
[math]\displaystyle{ n=0 }[/math] | 1 | ||||||||||||||||
[math]\displaystyle{ n=1 }[/math] | 1 | 1 | |||||||||||||||
[math]\displaystyle{ n=2 }[/math] | 1 | 1 | 1 | ||||||||||||||
[math]\displaystyle{ n=3 }[/math] | 1 | 2 | 2 | 1 | |||||||||||||
[math]\displaystyle{ n=4 }[/math] | 1 | 3 | 6 | 3 | 1 | ||||||||||||
[math]\displaystyle{ n=5 }[/math] | 1 | 5 | 15 | 15 | 5 | 1 | |||||||||||
[math]\displaystyle{ n=6 }[/math] | 1 | 8 | 40 | 60 | 40 | 8 | 1 | ||||||||||
[math]\displaystyle{ n=7 }[/math] | 1 | 13 | 104 | 260 | 260 | 104 | 13 | 1 |
The recurrence relation
- [math]\displaystyle{ \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 coefficients and the golden ratio [math]\displaystyle{ \varphi=\frac{1+\sqrt5}2 }[/math]:
- [math]\displaystyle{ {\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 Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence [math]\displaystyle{ G_n }[/math], that is, a sequence that satisfies [math]\displaystyle{ G_n = G_{n-1} + G_{n-2} }[/math] for every [math]\displaystyle{ n, }[/math] then
- [math]\displaystyle{ \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]\displaystyle{ n }[/math], and every nonnegative integer [math]\displaystyle{ k }[/math].
References
- Benjamin, Arthur T.; Plott, Sean S., A combinatorial approach to Fibonomial coefficients, Dept. of Mathematics, Harvey Mudd College, Claremont, CA 91711, http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf, retrieved 2009-04-04
- Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
- Weisstein, Eric W.. "Fibonomial Coefficient". http://mathworld.wolfram.com/FibonomialCoefficient.html.
- Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.
![]() | Original source: https://en.wikipedia.org/wiki/Fibonomial coefficient.
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