A152225 Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 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
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..2025
- Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Forests and pattern-avoiding permutations modulo pure descents, Permutation Patterns 2017, Reykjavik University, Iceland, June 26-30, 2017. See Section 5.
- Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
- 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, and Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289.
Crossrefs
Programs
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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 *)
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
Extensions
Edited by Emeric Deutsch, Dec 20 2008
Comments