A152172 a(n) is the number of Dyck paths of length n without height of peaks 0 (mod 3) and height of valleys 1 (mod 3).

1, 1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, 13054, 33598, 87143, 227542, 597640, 1577866, 4185108, 11146570, 29798682, 79932298, 215072896, 580327122, 1569942098, 4257254850, 11569980794, 31508150890, 85968266198, 234975421554

Offset

1,3

Links

Table of n, a(n) for n=1..31.

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 - 2*x^2 - sqrt(1 - 2*x - 3*x^2 + 4*x^4))/(2 - 2*x). - amended by Georg Fischer, Apr 09 2020

Conjecture: n*a(n) + 2*(1-n)*a(n-1) + (10-3*n)*a(n-2) + 4*a(n-3) + 4*(n-5)*a(n-4) = 0. - R. J. Mathar, Aug 14 2012

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

Mathematica

Rest[CoefficientList[Series[(1+x-2x^2-Sqrt[1-2x-3x^2+4x^4])/(2-2x), {x, 0, 30}], x]] (* Harvey P. Dale, Apr 10 2012; modified by Georg Fischer, Apr 09 2020 *)

Crossrefs

Cf. A086625. - R. J. Mathar, Dec 03 2008

Sequence in context: A166296 A151527 A086625 * A001677 A339150 A024422

Adjacent sequences: A152169 A152170 A152171 * A152173 A152174 A152175

Keyword

nonn

Author

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

Status

approved