This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372949 #34 Jun 27 2024 17:26:49 %S A372949 1,2,8,91,2618,199716,39690618,20689636692,28215085220016, %T A372949 100763710906257557,942012688139052139766,23056957423045790791793932, %U A372949 1477460537993359748548214768630,247860656992078740305125996374953260,108861324945456389643061592667638024842480 %N A372949 a(n) = 2*f(2*n)/(f(n)*f(n+2)) where f = A003266. %C A372949 Fibonacci analog of the super ballot numbers. %C A372949 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. %H A372949 Ira M. Gessel, <a href="https://doi.org/10.1016/0747-7171(92)90034-2">Super ballot numbers</a>, J. Symb. Comput. 14 (1992), 179-194. %H A372949 Kendra Killpatrick, <a href="https://arxiv.org/abs/2308.13457">Super FiboCatalan Numbers and their Lucas Analogues</a>, arXiv:2308.13457 [math.CO], (2023). %F A372949 a(n) ~ 10 * phi^((n-3)*(n+1)) / A062073, where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, May 29 2024 %e A372949 a(5) = 2*f(10)/(f(5)*f(7)) = 2*122522400/(30*3120) = 2618, where f=A003266. %t A372949 a[n_]:=2Fibonorial[2n]/(Fibonorial[n]Fibonorial[n+2]); Array[a,15] (* _Stefano Spezia_, May 23 2024 *) %Y A372949 Cf. A003266, A007054. %K A372949 nonn %O A372949 1,2 %A A372949 _Kendra Killpatrick_, May 17 2024