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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
OFFSET
0,4
LINKS
J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4.
G. Benkart and T. Halverson, Motzkin Algebras, Eur. J. Comb. 36 (2014) 473-502
Dennis E. Davenport, Louis W. Shapiro, and 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
D-finite with recurrence (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2) +(n+1)*a(n-3) +2*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Nov 13 2020
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 30 2005
STATUS
approved