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 A128721 Number of UUU's in all skew Dyck paths of semilength n. 2
 0, 0, 0, 4, 28, 157, 820, 4155, 20742, 102725, 506504, 2491230, 12236520, 60063399, 294748884, 1446436680, 7099442700, 34855583275, 171187439920, 841084246980, 4134129246180, 20328683526575, 100003531112300, 492153054177155 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203 FORMULA a(n) = Sum_{k=0..n-2} k*A128719(n,k) (n >= 2). G.f.: (2zg - g + 1 - z + z^2)/(2zg + z - 1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z). Recurrence: 2*(n+1)*(121*n-348)*a(n) = (1663*n^2 - 4620*n + 1392)*a(n-1) - (2476*n^2 - 11133*n + 11787)*a(n-2) + 5*(n-4)*(211*n-537)*a(n-3). - Vaclav Kotesovec, Nov 19 2012 a(n) ~ 9*5^(n-3/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 19 2012 EXAMPLE a(3)=4 because each of the paths UUUDDD, UUUDLD, UUUDDL and UUUDLL contains one UUU, while the other six paths of semilength 3 contain no UUU's. MAPLE G:=(1-5*z+4*z^2-2*z^3-(1-2*z)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 28): seq(coeff(Gser, z, n), n=0..25); MATHEMATICA CoefficientList[Series[(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)-(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+1-x+x^2)/(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2012 *) CROSSREFS Cf. A128719. Sequence in context: A123520 A012847 A273431 * A273647 A272936 A053524 Adjacent sequences: A128718 A128719 A128720 * A128722 A128723 A128724 KEYWORD nonn AUTHOR Emeric Deutsch, Mar 30 2007 STATUS approved

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Last modified August 13 05:07 EDT 2024. Contains 375113 sequences. (Running on oeis4.)