A152173 a(n) is the number of Dyck paths of length n without height of peaks 1 (mod 3) and height of valleys 2 (mod 3).

1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716, 14828, 38717, 101682, 268416, 711810, 1895432, 5066030, 13586082, 36547534, 98593064, 266661162, 722953814, 1964358938, 5348367006, 14589803090, 39870312218, 109136843138

Offset: 2

Author

Jun Ma (majun(AT)math.sinica.edu.tw), Nov 27 2008

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..1000
• Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289.

Crossrefs

Cf. A127389. - R. J. Mathar, Dec 03 2008

Programs

Mathematica

CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2+4x^4])/(2x^2 (1+x)),{x,0,30}],x] (* Harvey P. Dale, Feb 10 2015 *)

Formula

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

Conjecture: -n*a(n) + (n-3)*a(n-1) + (5*n-12)*a(n-2) + 3*(n-3)*a(n-3) + 4*(6-n)*a(n-4) + 4*(6-n)*a(n-5) = 0. - R. J. Mathar, Aug 14 2012

G.f.: 1/x^2 - 2/x + 2/(1+x) + G(0)/x where G(k) = 1 - 1/(x + x^2/(1 + x/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 28 2012