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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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) x='x+O('x^50); Vec(1/(1-x*(1-sqrt(1-4*x))/(2*x*(1+x^2))) \\ G. C. Greubel, Mar 17 2017

CROSSREFS

Cf. A114499.

Sequence in context: A255432 A275539 A193049 * A148212 A139627 A149844

Adjacent sequences:  A114497 A114498 A114499 * A114501 A114502 A114503

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 04 2005

STATUS

approved

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Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)