Fibonomial coefficient

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

${(\genfrac{}{}{0ex}{}{n}{k})}_F=\frac{F_n{F}_{n-1}\cdots F_{n-k+1}}{F_k{F}_{k-1}\cdots F_1}=\frac{n{!}_F}{k{!}_F(n-k){!}_F}$

where n and k are non-negative integers, 0 ≤ k ≤ n, F_j 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:

${(\genfrac{}{}{0ex}{}{n}{0})}_F={(\genfrac{}{}{0ex}{}{n}{n})}_F=1$

${(\genfrac{}{}{0ex}{}{n}{1})}_F={(\genfrac{}{}{0ex}{}{n}{n-1})}_F=F_n$

${(\genfrac{}{}{0ex}{}{n}{2})}_F={(\genfrac{}{}{0ex}{}{n}{n-2})}_F=\frac{F_n{F}_{n-1}}{F_2{F}_1}=F_n{F}_{n-1},$

${(\genfrac{}{}{0ex}{}{n}{3})}_F={(\genfrac{}{}{0ex}{}{n}{n-3})}_F=\frac{F_n{F}_{n-1}{F}_{n-2}}{F_3{F}_2{F}_1}=F_n{F}_{n-1}{F}_{n-2}/2,$

${(\genfrac{}{}{0ex}{}{n}{k})}_F={(\genfrac{}{}{0ex}{}{n}{n-k})}_F.$

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.

$n=0$									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

${(\genfrac{}{}{0ex}{}{n}{k})}_F=F_{n-k+1}{(\genfrac{}{}{0ex}{}{n-1}{k-1})}_F+F_{k-1}{(\genfrac{}{}{0ex}{}{n-1}{k})}_F$

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

${(\genfrac{}{}{0ex}{}{n}{k})}_F={\phi }^{k(n-k)}{(\genfrac{}{}{0ex}{}{n}{k})}_{-1/{\phi }^2}$

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

$\sum _{j=0}^{k+1}(-1{)}^{j(j+1)/2}{(\genfrac{}{}{0ex}{}{k+1}{j})}_F{G}_{n-j}^k=0,$

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

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.