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Search: id:A128721
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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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+0
2
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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; listen)
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OFFSET
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0,4
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COMMENT
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a(n)=Sum(k*A128719(n,k), k=0..n-2) (n>=2).
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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.=(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
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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
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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
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Cf. A128719.
Sequence in context: A006302 A123520 A012847 this_sequence A053524 A125687 A026298
Adjacent sequences: A128718 A128719 A128720 this_sequence A128722 A128723 A128724
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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 30 2007
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