A128723 Number of skew Dyck paths of semilength n having no peaks at level 1.

1, 0, 2, 6, 22, 84, 334, 1368, 5734, 24480, 106086, 465462, 2063658, 9231084, 41610162, 188820726, 861891478, 3954732384, 18230522422, 84390187986, 392120098258, 1828220666844, 8550445900442, 40103716079436

Offset

0,3

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Helmut Prodinger, Skew Dyck paths having no peaks at level 1, arXiv:2201.00640 [math.CO], 2022.

Formula

a(n) = A128722(n,0).

a(n) = 2*A117641(n) for n>=1.

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

a(n) ~ 5^(n+3/2)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: +3*(n+1)*a(n) +(-17*n+10)*a(n-1) +9*(n-3)*a(n-2) +5*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

Example

a(3)=6 because we have UUDUDD, UUUDDD, UUUDLD, UUDUDL, UUUDDL and UUUDLL.

Maple

G:=(3-3*z-sqrt(1-6*z+5*z^2))/(1+3*z+sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);

Mathematica

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

Prog

(PARI) my(z='x+O('z^50)); Vec((3-3*z-sqrt(1-6*z+5*z^2)) /(1+3*z +sqrt(1-6*z+5*z^2))) \\ G. C. Greubel, Mar 19 2017

Crossrefs

Cf. A117641, A128722.

Sequence in context: A164870 A121686 A245904 * A150244 A151288 A150245

Adjacent sequences: A128720 A128721 A128722 * A128724 A128725 A128726

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved