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Search: id:A128714
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| A128714 |
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Number of skew Dyck paths of semilength n ending with a left step. 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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3
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| 0, 0, 1, 4, 15, 58, 232, 954, 4010, 17156, 74469, 327168, 1452075, 6501156, 29326743, 133166064, 608188737, 2791992736, 12876049123, 59626721244, 277150709717, 1292583258866, 6046985696778, 28369001791034, 133436435891480
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of skew Dyck paths of semilength n and ending with a down step is A033321(n).
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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.=[1-3z-sqrt(1-6z+5z^2)]/[1+z+sqrt(1-6z+5z^2)]. G.f.=z(g-1)/(1-zg), 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 we have UDUUDL, UUDUDL, UUUDDL, and UUUDLL.
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MAPLE
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G:=(1-3*z-sqrt(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..27);
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CROSSREFS
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Cf. A033321.
Sequence in context: A017950 A003126 A102052 this_sequence A007342 A017951 A129155
Adjacent sequences: A128711 A128712 A128713 this_sequence A128715 A128716 A128717
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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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