OFFSET
0,4
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)
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
MAPLE
b:= proc(x, y, h, z) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, `if`(h, 0, 1),
b(x-1, y, h, is(x=y))+ `if`(h and z, b(x, y-1, false$2),
`if`(z, 0, b(x, y-1, h, false)))))
end:
a:= n-> 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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 02 2001
EXTENSIONS
More terms from Emeric Deutsch, Dec 03 2001
STATUS
approved