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 A128750 Number of skew Dyck paths of semilength n having no ascents of length 1. 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. An ascent in a path is a maximal sequence of consecutive U steps. 2
 1, 0, 2, 4, 14, 44, 150, 520, 1850, 6696, 24602, 91500, 343846, 1303572, 4979822, 19150352, 74075890, 288022160, 1125076210, 4413061972, 17375007294, 68641377980, 272014578822, 1081009104664, 4307221752874, 17203123381304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n)=A128749(n,0). Hankel transform is 2^ceiling(n(n+1)/3). Binomial transform is A059278. [From Paul Barry, Feb 11 2009] LINKS E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203 FORMULA G.f.=G=G(z) satisfies z(1+z)G^2-(1-z^2)G+1-z=0. G.f.: 1/(1+x-x/(1-x-x/(1+x-x/(1-x-x/(1+x-x/(1-... (continued fraction). [From Paul Barry, Feb 11 2009] Contribution from Paul Barry, Feb 11 2009: (Start) G.f.: (1/(1+x))c(x/(1-x^2)) where c(x) is the g.f. of A000108; G.f.: 1/(1-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-.... (continued fraction); a(n)=sum{k=0..n, (-1)^(n-k)*C(floor((n+k)/2),k)*A000108(k)}. (End) Conjecture: (n+1)*a(n) +(-4*n+3)*a(n-1) +(-2*n-1)*a(n-2) +(4*n-11)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2012 EXAMPLE a(3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL. MAPLE G:=(1-z^2-sqrt((1-z^2)*(1-4*z-z^2)))/2/z/(1+z): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..30); CROSSREFS Cf. A128749. Sequence in context: A152011 A000912 A169982 * A047152 A007866 A121751 Adjacent sequences:  A128747 A128748 A128749 * A128751 A128752 A128753 KEYWORD nonn AUTHOR Emeric Deutsch, Mar 31 2007 STATUS approved

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