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

0,2

Comments

Hankel transform is A134751. Binomial transform of A105864. [From Paul Barry, Oct 07 2008]

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.

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

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 *)

Prog

(PARI) Vec((1 - sqrt(1 - 4*t*(1 - t)/(1 - 2*t)))/(2*t) + O(t^50)) \\ G. C. Greubel, Jan 04 2017

Crossrefs

Sequence in context: A338184 A348351 A150134 * A154381 A150135 A150136

Adjacent sequences: A059276 A059277 A059278 * A059280 A059281 A059282

Keyword

nonn

Author

N. J. A. Sloane, Jan 24 2001

Status

approved