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A128732
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Number DL's in all skew Dyck paths of semilength n. 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 the path is defined to be the number of its steps.
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1
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0, 0, 1, 5, 23, 106, 493, 2312, 10917, 51840, 247319, 1184557, 5692517, 27434578, 132547877, 641789941, 3113487683, 15130119784, 73637665027, 358883327591, 1751237017413, 8555108199294, 41836182269267, 204779733440086
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=Sum(k*A128731(n,k), k>=0).
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| G.f.=z[1-z-sqrt(1-6z+5z^2)]/[1-6z+5z^2 +(1+z)sqrt(1-6z+5z^2)].
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EXAMPLE
| a(3)=5 because we have UDUUDL, UUUDLD, UUDUDL, UUUDDL and UUUDLL (the remaining 5 paths are Dyck paths which, obviously, contain no DL's).
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MAPLE
| G:=z*(1-z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..26);
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CROSSREFS
| Cf. A128731.
Sequence in context: A026760 A064914 A107839 * A026894 A126473 A109877
Adjacent sequences: A128729 A128730 A128731 * A128733 A128734 A128735
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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