A135336 Number of Dyck paths of semilength n with no UUDU's starting at level 0.

1, 1, 2, 4, 10, 28, 85, 271, 893, 3013, 10351, 36075, 127219, 453097, 1627378, 5887660, 21436354, 78484402, 288779728, 1067263660, 3960081904, 14746806292, 55094725918, 206450572930, 775724723086, 2922060848734

Offset

0,3

Comments

Column 0 of A135330. Partial sums of the Fine sequence 1,0,1,2,6,18,... (A000957 without the first term). - Emeric Deutsch, Dec 14 2007

Links

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

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Formula

From Emeric Deutsch, Dec 14 2007: (Start)

a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(3*j+1)*binomial(2*n-3*j,n)/(n+1).

G.f.: C/(1+z^3*C^3) = C/[(1-z)*(1+z*C)], where C = [1-sqrt(1-4*z)]/(2*z) is the g.f. of the Catalan numbers (A000108). (End)

a(n) ~ 4^(n+2)/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Example

a(3)=4 because among the 5 (=A000108(3)) Dyck paths of semilength 3 only UUDUDD does not qualify.

Maple

a:=proc(n) options operator, arrow: (sum((-1)^j*(3*j+1)*binomial(2*n-3*j, n), j =0..floor((1/3)*n)))/(n+1) end proc: seq(a(n), n=0..25); # Emeric Deutsch, Dec 14 2007

Mathematica

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

Prog

(PARI) x='x+O('x^50); Vec((1-sqrt(1-4*x))/(x*(1-x)*(3 - sqrt(1-4*x)))) \\ G. C. Greubel, Mar 21 2017

Crossrefs

Cf. A000108, A135330, A000957.

Sequence in context: A348197 A289709 A192574 * A149825 A149826 A149827

Adjacent sequences: A135333 A135334 A135335 * A135337 A135338 A135339

Keyword

nonn

Author

N. J. A. Sloane, Dec 07 2007

Extensions

Edited and extended by Emeric Deutsch, Dec 14 2007

Status

approved