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A128721
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Number of UUU'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 a path is defined to be the number of its steps.
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1
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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; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=Sum(k*A128719(n,k), k=0..n-2) (n>=2).
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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.=(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).
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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.
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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);
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CROSSREFS
| Cf. A128719.
Sequence in context: A006302 A123520 A012847 * A053524 A125687 A026298
Adjacent sequences: A128718 A128719 A128720 * A128722 A128723 A128724
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2007
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