OFFSET
0,4
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.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
FORMULA
G.f.: 2*(1+x)/(1+x+2*x^2+(1+x)*sqrt(1-4*x)).
a(n) ~ 25*4^(n+1)/(121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: (n+1)*a(n) +3*a(n-1) +(-11*n+16)*a(n-2) +(-17*n+43)*a(n-3) +(-11*n+28)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jun 17 2016
EXAMPLE
a(3)=4 because we have UDUUDD, UUDDUD, UUDUDD and UUUDDD.
a(4)=11 because among the 14 (=A000108(4)) Dyck paths of semilength 4 the paths that do not qualify are UDUDUDUD, UDUDUUDD and UUDDUDUD.
MAPLE
g:=2*(1+z)/(1+z+2*z^2+sqrt(1-4*z)+z*sqrt(1-4*z)): gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=0..30);
MATHEMATICA
CoefficientList[Series[2*(1+x)/(1+x+2*x^2+Sqrt[1-4*x]+x*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) x='x+O('x^50); Vec(2*(1+x)/(1+x+2*x^2+(1+x)*sqrt(1-4*x))) \\ G. C. Greubel, Mar 19 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 27 2007, Dec 13 2007
STATUS
approved