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Fibonomial Coefficient

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The fibonomial coefficient (sometimes also called simply the Fibonacci coefficient) is defined by

[m; k]_F=(F_mF_(m-1)...F_(m-k+1))/(F_1F_2...F_k),
(1)

where [m; 0]_F=1 and F_n is a Fibonacci number. This coefficient satisfies

[m; n]_F=1/2(L_n[m-1; n]_F+L_(m-n)[m-1; n-1]_F)
(2)

for k>0, where L_n is a Lucas number.

The triangle of fibonomial coefficients is given by

1 1,1 1,1,1 1,2,2,1 1,3,6,3,1 1,5,15,15,5,1
(3)

(OEIS A010048).

[2n; n] may be called the central fibonomial coefficient by analogy with the central binomial coefficient.

See also

Binomial Coefficient, Central Fibonomial Coefficient, Fibonacci Number, Fibonorial, Lucas Number

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References

Benjamin, A. T. and Quinn, J. J. Proofs That Really Count: the Art of Combinatorial Proof. Washington, DC: Math. Assoc. Amer., p. 15, 2003.Brousseau, A. Fibonacci and Related Number Theoretic Tables. San Jose, CA: Fibonacci Association, 1972.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 84 and 492, 1997.Krot, E. "Further Developments in Finite Fibonomial Calculus." 27 Oct 2004. http://arxiv.org/abs/math.CO/0410550.Richardson, T. M. "The Filbert Matrix." 12 May 1999. http://arxiv.org/abs/math/9905079.Sloane, N. J. A. Sequence A010048 in "The On-Line Encyclopedia of Integer Sequences."

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Fibonomial Coefficient

Cite this as:

Weisstein, Eric W. "Fibonomial Coefficient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonomialCoefficient.html

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