A292460 Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3) in powers of x.

1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374

Offset: 0

Author

Seiichi Manyama, Sep 16 2017

Keywords

• nonn

Comments

Number of U_{k}D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - Sergey Kirgizov, Apr 08 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
• Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

Crossrefs

Cf. A001006, A004148, A292461.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 -Sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3))); // G. C. Greubel, Aug 13 2018

Mathematica

CoefficientList[Series[(1-x-x^2 -Sqrt[(1-x-x^2)^2 -4*x^3])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)

PARI

x='x+O('x^50); Vec((1-x-x^2 -sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3)) \\ G. C. Greubel, Aug 13 2018

Formula

G.f.: 1/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction).

a(n) = A004148(n+1).

a(n) ~ 5^(1/4) * phi^(2*n + 4) / (2*sqrt(Pi)*n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Sep 17 2017

D-finite with recurrence: (n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jan 23 2020

a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Nov 09 2021