A065601 Number of Dyck paths of length 2n with exactly 1 hill.

0, 1, 0, 2, 4, 13, 40, 130, 432, 1466, 5056, 17672, 62460, 222853, 801592, 2903626, 10582816, 38781310, 142805056, 528134764, 1960825672, 7305767602, 27307800400, 102371942932, 384806950624, 1450038737668, 5476570993440, 20727983587220, 78606637060012

Offset: 0

Author

N. J. A. Sloane, Dec 02 2001

Keywords

• nonn
• easy

Comments

Convolution of A000957(n) with itself gives a(n-1).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
• E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
• E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012
• Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)

Crossrefs

2nd column of A065600. Cf. A000957.

Programs

Maple

b:= proc(x, y, h, z) option remember;
     `if`(x<0 or y b(n$2, true$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2012
# second Maple program:
series(((1-sqrt(1-4*x))/(3-sqrt(1-4*x)))^2/x, x=0, 30);  # Mark van Hoeij, Apr 18 2013

Mathematica

CoefficientList[Series[((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))^2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
Table[Sum[(-1)^j*(j+1)*(j+2)*Binomial[2*n-1-j,n],{j,0,n-1}]/(n+1),{n,0,30}] (* Vaclav Kotesovec, May 18 2015 *)

Formula

Reference gives g.f.'s.

Conjecture: 2*(n+1)*a(n) +(-3*n+2)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Dec 10 2013

a(n) ~ 2^(2*n+3) / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

Extensions

More terms from Emeric Deutsch, Dec 03 2001