A129155 Number of skew Dyck paths of semilength n that have no primitive Dyck factors.

1, 0, 1, 4, 15, 59, 241, 1011, 4326, 18797, 82685, 367410, 1646494, 7432270, 33761322, 154213566, 707882503, 3263713148, 15107319268, 70182332975, 327111450097, 1529226524057, 7168880978609, 33693179852563

Offset: 0

Author

Emeric Deutsch, Apr 02 2007

Keywords

• nonn

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.

Examples

			a(3)=4 because we have UUUDLD, UUDUDL, UUUDDL and UUUDLL.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Crossrefs

Cf. A129154, A129157.

Programs

Maple

G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);

Mathematica

CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])/(2+x-Sqrt[1-4*x]+Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PARI

z='z+O('z^50); Vec((3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2))) \\ G. C. Greubel, Mar 20 2017

Formula

a(n) = A129154(n,0).

G.f.: (3-3*z-sqrt(1-6*z+5*z^2))/(2+z-sqrt(1-4*z)+sqrt(1-6*z+5*z^2)).

a(n) ~ (475 + 697*sqrt(5)) * 5^n / (3364*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014