A128732 Number DL's in all skew Dyck paths of semilength n.

0, 0, 1, 5, 23, 106, 493, 2312, 10917, 51840, 247319, 1184557, 5692517, 27434578, 132547877, 641789941, 3113487683, 15130119784, 73637665027, 358883327591, 1751237017413, 8555108199294, 41836182269267, 204779733440086

Offset

0,4

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.

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

Formula

a(n) = Sum_{k>=0} k*A128731(n,k).

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

a(n) ~ 5^(n-1/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: +2*n*(3*n-1)*a(n) -n*(39*n-37)*a(n-1) +4*(12*n^2-22*n-15)*a(n-2) -5*(3*n+2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

Example

a(3)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the remaining 5 paths are Dyck paths which, obviously, contain no DL's).

Maple

G:=z*(1-z-sqrt(1-6*z+5*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[x*(1-x-Sqrt[1-6*x+5*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 *)

Prog

(PARI) z='z+O('z^50); concat([0, 0], Vec(z*(1-z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2 +(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017

Crossrefs

Cf. A128731.

Sequence in context: A239406 A107839 A270530 * A026894 A126473 A238112

Adjacent sequences: A128729 A128730 A128731 * A128733 A128734 A128735

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved