A128730 Number of UDL's in all skew Dyck paths of semilength n.

0, 0, 1, 4, 16, 68, 301, 1366, 6301, 29400, 138355, 655424, 3121438, 14930540, 71675839, 345148892, 1666432816, 8064278288, 39103576699, 189949958332, 924163714216, 4502711570988, 21966152501239, 107284324830302

Offset: 0

Author

Emeric Deutsch, Mar 31 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 a path is defined to be the number of steps in it.

Examples

			a(3) = 4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL (the other six skew Dyck paths of semilength 3 are the five Dyck paths and UUUDDL).
		

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. A128728, A014300.

Programs

Maple

G:=2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..26);

Mathematica

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

PARI

z='z+O('z^50); concat([0,0], Vec(2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 19 2017

Formula

a(n) = Sum_{k>=0} k*A128728(n,k).

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

a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence: +2*(n-1)*(3*n-8)*a(n) +(-39*n^2+161*n-148)*a(n-1) +(48*n^2-215*n+220)*a(n-2) -5*(3*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

For n >= 2, a(n) = Sum_{k=1..n-1} binomial(n,k)*A014300(k). - Jianing Song, Apr 20 2019