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A152225 Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 3). 3
1, 1, 2, 4, 9, 22, 56, 146, 388, 1048, 2869, 7942, 22192, 62510, 177308, 506008, 1451866, 4185788, 12119696, 35227748, 102753800, 300672368, 882373261, 2596389190, 7658677856, 22642421206, 67081765932, 199128719896, 592179010350, 1764044315540, 5263275015120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The antidiagonal sums of A091894 equal this sequence. - Johannes W. Meijer, Sep 13 2012

LINKS

Robert Israel, Table of n, a(n) for n = 0..2025

Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Forests and pattern-avoiding permutations modulo pure descents, Permutation Patterns 2017, Reykjavik University, Iceland, June 26-30, 2017. See Section 5.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289.

FORMULA

G.f.: (1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3))/(2*x^2).

Conjecture: (n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Aug 14 2012

This conjecture follows from the differential equation (4*x^4-4*x^3+4*x^2-x)*y' + (2*x^3-4*x^2+6*x-2)*y - 2*x^3+2*x^2-3*x+2=0 satisfied by the g.f. - Robert Israel, Jan 09 2018

MAPLE

f:= gfun:-rectoproc({(n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0, a(0)=1, a(1)=1, a(2)=2, a(3)=4}, a(n), remember):

map(f, [$0..40]); # Robert Israel, Jan 09 2018

MATHEMATICA

CoefficientList[Series[(1 - 2 x + 2 x^2 - Sqrt[1 - 4 x + 4 x^2 - 4 x^3])/(2 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Jan 09 2018 *)

CROSSREFS

Cf. A091561, A025265, A025247. - R. J. Mathar, Dec 03 2008

Sequence in context: A088456 A091561 A025265 * A037245 A244886 A143017

Adjacent sequences:  A152222 A152223 A152224 * A152226 A152227 A152228

KEYWORD

nonn

AUTHOR

Majun (majun(AT)math.sinica.edu.tw), Nov 29 2008

EXTENSIONS

Edited by Emeric Deutsch, Dec 20 2008

STATUS

approved

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Last modified February 20 18:39 EST 2019. Contains 320345 sequences. (Running on oeis4.)