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Central Fibonomial Coefficient

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The nth central fibonomial coefficient, for a positive integer n, is defined as

[2n; n]_F=product_(k=1)^(n)(F_(n+k))/(F_k)
(1)
=-(phi^(n^2)((-1)^nphi^(-2n);-phi^(-2))_(n+1))/([(-1)^nphi^(-2n)-1](-phi^(-2);-phi^(-2))_n),
(2)

where [n; k]_F is a fibonomial coefficient, F_n is a Fibonacci number, phi is the golden ratio, and (a;q)_n is a q-Pochhammer symbol (E. W. Weisstein, Dec. 8, 2009). At n=0, the defining empty product has value 1.

For n=1, 2, ..., the first few are 1, 6, 60, 1820, 136136, ... (OEIS A003267).

See also

Central Binomial Coefficient, Central Trinomial Coefficient, Fibonomial Coefficient

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References

Brousseau, A. "A Sequence of Power Formulas." Fib. Quart. 6, 81-83, 1968. https://www.fq.math.ca/Scanned/6-1/brousseau3.pdf.Brousseau, A. Fibonacci and Related Number Theoretic Tables. Santa Clara, CA: Fibonacci Association, p. 74, 1972. https://www.fq.math.ca/fibonacci-tables.html.Sloane, N. J. A. Sequence A003267/M4272 in "The On-Line Encyclopedia of Integer Sequences."

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Central Fibonomial Coefficient

Cite this as:

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

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