Fibonomial coefficient

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

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[edit]

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

Fibonomial triangle[edit]

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.

1
1 1
1 1 1
1 2 2 1
1 3 6 3 1
1 5 15 15 5 1
1 8 40 60 40 8 1
1 13 104 260 260 104 13 1

The recurrence relation

implies that the Fibonomial coefficients are always integers.

References[edit]