A114500 Number of Dyck paths of semilength n having no UUUDDD's starting at level zero; here U=(1,1), D=(1,-1). Also number of Dyck paths of semilength n having no UUDUDD's starting at level zero.

1, 1, 2, 4, 12, 37, 119, 390, 1307, 4460, 15452, 54207, 192170, 687386, 2477810, 8992007, 32825653, 120460613, 444125661, 1644324767, 6111002752, 22789116600, 85251100275, 319826371389, 1203008722282, 4536009027311, 17141555233270

Offset

0,3

Comments

Column 0 of A114499.

Links

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

Formula

G.f.: 1/(1 - z*C + z^3), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.

a(n) ~ 4^(n+5)/(1089*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence +(n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-2) +2*(-2*n+1)*a(n-3) +(n+1)*a(n-5) +2*(-2*n+1)*a(n-6)=0. - R. J. Mathar, Jul 26 2022

Example

a(4)=12 because among the 14 Dyck paths of semilength 4 only UDUUUDDD and UUUDDDUD contain UUUDDD starting at level 0.

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..30);

Mathematica

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

Prog

(PARI) my(x='x+O('x^50)); Vec(2/(1+sqrt(1-4*x)+2*x^3)) \\ Jason Yuen, Sep 09 2024

Crossrefs

Cf. A114499.

Sequence in context: A275539 A356062 A193049 * A148212 A139627 A361720

Adjacent sequences: A114497 A114498 A114499 * A114501 A114502 A114503

Keyword

nonn,changed

Author

Emeric Deutsch, Dec 04 2005

Status

approved