A128736 Number of skew Dyck paths of semilength n and having no LDU's.

1, 1, 3, 10, 35, 127, 474, 1810, 7043, 27839, 111503, 451640, 1847032, 7616692, 31637664, 132252886, 555967283, 2348920207, 9968617393, 42477135370, 181661283779, 779492777031, 3354893322350, 14479454240492, 62652100034380

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 the path is defined to be the number of its steps.

Links

Table of n, a(n) for n=0..24.

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

Formula

a(n) = A128735(n,0).

G.f.: G = G(z) satisfies zG^3 = (1-2z)(G-1)(2G-1).

D-finite with recurrence 8*n*(n+1)*a(n) -4*n*(11*n-2)*a(n-1) +2*(-11*n^2+114*n-154)*a(n-2) +(61*n-153)*(5*n-16)*a(n-3) -4*(47*n-159)*(n-4)*a(n-4) -220*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Jul 22 2022

Example

a(4)=35 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.

Maple

eq:=z*G^3=(1-2*z)*(G-1)*(2*G-1): G:=RootOf(eq, G): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);

Crossrefs

Cf. A128735.

Sequence in context: A346739 A078789 A299443 * A303730 A149037 A228769

Adjacent sequences: A128733 A128734 A128735 * A128737 A128738 A128739

Keyword

nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved