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A372949

a(n) = 2*f(2*n)/(f(n)*f(n+2)) where f = A003266.

1, 2, 8, 91, 2618, 199716, 39690618, 20689636692, 28215085220016, 100763710906257557, 942012688139052139766, 23056957423045790791793932, 1477460537993359748548214768630, 247860656992078740305125996374953260, 108861324945456389643061592667638024842480

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OFFSET

1,2

COMMENTS

Fibonacci analog of the super ballot numbers.

a(n) is also the generalized FiboCatalan number for r=1. Proof that the formula always gives a positive integer can be found in a recent paper of K. Killpatrick. The sequence is the Fibonacci analog of the super ballot numbers given by Gessel (A007054). The sequence is also the Fibonacci analog of the generalized Catalan numbers, J_r*(2n)!/(n!*(n+r+1)!) where J_r=(2r+1)!/r!, for r=1. Gessel defined the generalized Catalan numbers and proved they are integers.

LINKS

Table of n, a(n) for n=1..15.

Ira M. Gessel, Super ballot numbers, J. Symb. Comput. 14 (1992), 179-194.

Kendra Killpatrick, Super FiboCatalan Numbers and their Lucas Analogues, arXiv:2308.13457 [math.CO], (2023).

FORMULA

a(n) ~ 10 * phi^((n-3)*(n+1)) / A062073, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2024

EXAMPLE

a(5) = 2*f(10)/(f(5)*f(7)) = 2*122522400/(30*3120) = 2618, where f=A003266.

MATHEMATICA

a[n_]:=2Fibonorial[2n]/(Fibonorial[n]Fibonorial[n+2]); Array[a, 15] (* Stefano Spezia, May 23 2024 *)

CROSSREFS

Cf. A003266, A007054.