A059279 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.
1, 2, 6, 20, 72, 276, 1112, 4656, 20080, 88608, 398144, 1815248, 8375904, 39037120, 183493440, 868853120, 4140414720, 19841656960, 95559048960, 462268075520, 2245165391360, 10943794652160, 53519094753280, 262510076263680, 1291131867203072
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 18.
Programs
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Mathematica
CoefficientList[Series[(1 - Sqrt[1 - 4*t*(1 - t)/(1 - 2*t)])/(2*t), {t, 0, 50}], t] (* G. C. Greubel, Jan 04 2017 *) -
PARI
Vec((1 - sqrt(1 - 4*t*(1 - t)/(1 - 2*t)))/(2*t) + O(t^50)) \\ G. C. Greubel, Jan 04 2017
Formula
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
G.f.: (1 - sqrt(1 - 4*x*(1 - x)/(1 - 2*x)))/(2*x). - G. C. Greubel, Jan 04 2017
G.f. A(x) satisfies: A(x) = 1 + x * (1/(1 - 2*x) + A(x)^2). - Ilya Gutkovskiy, Jun 30 2020
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
Comments