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

2,4

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.

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

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 *)

Crossrefs

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

Sequence in context: A366229 A137681 A127389 * A032171 A127713 A354076

Adjacent sequences: A152170 A152171 A152172 * A152174 A152175 A152176

Keyword

nonn

Author

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

Status

approved