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Fibonomial coefficient

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

Contents

( n k ) F = F n F n − 1 ⋯ F n − k + 1 F k F k − 1 ⋯ F 1 = 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:

( n 0 ) F = ( n n ) F = 1 ( n 1 ) F = ( n n − 1 ) F = F n ( n 2 ) F = ( n n − 2 ) F = F n F n − 1 F 2 F 1 = F n F n − 1 , ( n 3 ) F = ( n n − 3 ) F = F n F n − 1 F n − 2 F 3 F 2 F 1 = F n F n − 1 F n − 2 / 2 , ( n k ) F = ( 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.

The recurrence relation

( n k ) F = F n − k + 1 ( n − 1 k − 1 ) F + F k − 1 ( 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 ϕ = 1 + 5 2 :

( n k ) F = ϕ k ( n − k ) ( n k ) − 1 / ϕ 2

References

Fibonomial coefficient Wikipedia
(Text) CC BY-SA


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