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

Offset: 1

Author

Kendra Killpatrick, May 17 2024

Keywords

• nonn

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.

Examples

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

Links

• 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).

Crossrefs

Cf. A003266, A007054.

Programs

Mathematica

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

Formula

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