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A114487 Number of Dyck paths of semilength n having no UUDD's starting at level 0. 2
1, 1, 1, 3, 10, 31, 98, 321, 1078, 3686, 12789, 44919, 159407, 570704, 2058817, 7476621, 27310345, 100275628, 369886451, 1370066394, 5093778398, 19002602171, 71109895075, 266855940177, 1004045604976, 3786790901401, 14313706230574, 54215799080454 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science,  Vol 17, No 3 (2016). See Table 4.

Dennis E. Davenport, Louis W. Shapiro, Leon C. Woodson, A bijection between the triangulations of convex polygons and ordered trees, Integers (2020) Vol. 20, Article #A8.

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

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

FORMULA

G.f.: 2/(1+2*z^2+sqrt(1-4*z)).

a(n) ~ 4^(n+3) / (81*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

a(n) = Sum_{k=0..n/2} (-1)^k*(k+1)/(2*n-3*k+1)*binomial(2*n-3*k+1, n-2*k). - Ira M. Gessel, Jun 16 2018

EXAMPLE

a(3) = 3 because we have UDUDUD, UUDUDD and UUUDDD, where U=(1,1), D=(1,-1).

MAPLE

G:=2/(1+2*z^2+sqrt(1-4*z)): Gser:=series(G, z=0, 33): 1, seq(coeff(Gser, z^n), n=1..30);

MATHEMATICA

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

PROG

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

CROSSREFS

Column 0 of A114486.

Sequence in context: A100058 A002160 A214839 * A017934 A005510 A005725

Adjacent sequences:  A114484 A114485 A114486 * A114488 A114489 A114490

KEYWORD

nonn,changed

AUTHOR

Emeric Deutsch, Nov 30 2005

STATUS

approved

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)