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A128750
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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.
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2
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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
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OFFSET
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0,3
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COMMENTS
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a(n)=A128749(n,0).
Hankel transform is 2^ceiling(n(n+1)/3). Binomial transform is A059278. [From Paul Barry, Feb 11 2009]
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LINKS
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Table of n, a(n) for n=0..25.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
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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
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EXAMPLE
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a(3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
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MAPLE
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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);
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CROSSREFS
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Cf. A128749.
Sequence in context: A152011 A000912 A169982 * A047152 A007866 A121751
Adjacent sequences: A128747 A128748 A128749 * A128751 A128752 A128753
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Mar 31 2007
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STATUS
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approved
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