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Search: id:A128750
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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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+0
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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 (pbarry(AT)wit.ie), Feb 11 2009]
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REFERENCES
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E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
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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 (pbarry(AT)wit.ie), Feb 11 2009]
Contribution from Paul Barry (pbarry(AT)wit.ie), 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)
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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: A079995 A152011 A000912 this_sequence A047152 A007866 A121751
Adjacent sequences: A128747 A128748 A128749 this_sequence A128751 A128752 A128753
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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