Fibonomial Coefficient is Integer

Theorem

Let $\dbinom n k_\FF$ be a Fibonomial coefficient.

Then $\dbinom n k_\FF$ is an integer.

Proof

Recurrence Relation for Fibonomial Coefficients gives:

$\dbinom n k_\FF = F_{k - 1} \dbinom {n - 1} k_\FF + F_{n - k + 1} \dbinom {n - 1} {k - 1}_\FF$

where $F_{k - 1}$ etc. denote Fibonacci numbers.

It follows that each Fibonomial coefficient is the sum of integers, and so an integer.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $29 \ \text{(b)}$