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 A135336 Number of Dyck paths of semilength n with no UUDU's starting at level 0. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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: A262543 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

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