# Definition:Fibonomial Coefficient

## Definition

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the Fibonomial coefficient $\dbinom n k$ is defined as:

$\dbinom n k_\FF = \begin{cases} 0 & : n < 0, n > k \\ 1 & : n \ge 0, k = 0 \\ \dfrac {F_n F_{n - 1} \cdots F_{n - k + 1} } {F_k F_{k - 1} \cdots F_1} = \ds \prod_{j \mathop = 1}^k \dfrac {F_{n - k + j} } {F_j} & : \text{otherwise} \end{cases}$

where $F_n$ denotes the $n$th Fibonacci number.

## Also known as

Some sources use the more verbose but also more descriptive Fibonacci-binomial coefficient.

## Also see

• Results about Fibonomial coefficients can be found here.

## Source of Name

This entry was named for Leonardo Fibonacci.

## Historical Note

The Fibonomial coefficients were initially defined by François Édouard Anatole Lucas.