Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #24 Mar 03 2024 14:43:22
%S 1,2,6,20,72,276,1112,4656,20080,88608,398144,1815248,8375904,
%T 39037120,183493440,868853120,4140414720,19841656960,95559048960,
%U 462268075520,2245165391360,10943794652160,53519094753280,262510076263680,1291131867203072
%N G.f. is ((1-x)/(1-2*x)) * G(x*(1-x)/(1-2*x)) where G(x) is g.f. for Catalan numbers A000108.
%C Hankel transform is A134751. Binomial transform of A105864. [From _Paul Barry_, Oct 07 2008]
%H G. C. Greubel, <a href="/A059279/b059279.txt">Table of n, a(n) for n = 0..1000</a>
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 18.
%F Conjecture: (n+1)*a(n) +2*(1-4*n)*a(n-1) + 4*(4*n-5)*a(n-2) +4*(5-2*n)*a(n-3)=0. - _R. J. Mathar_, Nov 15 2011
%F G.f.: (1 - sqrt(1 - 4*x*(1 - x)/(1 - 2*x)))/(2*x). - _G. C. Greubel_, Jan 04 2017
%F G.f. A(x) satisfies: A(x) = 1 + x * (1/(1 - 2*x) + A(x)^2). - _Ilya Gutkovskiy_, Jun 30 2020
%F a(n) ~ 5^(1/4) * 2^(n-1) * phi^(2*n + 3/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 30 2020
%t CoefficientList[Series[(1 - Sqrt[1 - 4*t*(1 - t)/(1 - 2*t)])/(2*t), {t, 0, 50}], t] (* _G. C. Greubel_, Jan 04 2017 *)
%o (PARI) Vec((1 - sqrt(1 - 4*t*(1 - t)/(1 - 2*t)))/(2*t) + O(t^50)) \\ _G. C. Greubel_, Jan 04 2017
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 24 2001