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Search: id:A128726
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| A128726 |
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Number of LL'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 steps in it. |
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+0
2
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| 0, 0, 0, 1, 7, 38, 192, 946, 4616, 22440, 108964, 529133, 2571079, 12504038, 60872038, 296641049, 1447054867, 7065841496, 34534088328, 168933369259, 827073303197, 4052396628306, 19870029768028, 97495408609784
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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a(n)=Sum(k*A128724(n,k),k=0..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.=z(1-g+zg^2)/[(1-z-zg)(1+z-3zg)], 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(4)=7 because we have UDUUUDLL, UUUUDLLD, UUDUUDLL, UUUDUDLL, UUUUDDLL and UUUUDLLL (last path has two LL's).
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/(2*z): G:=z*(g-1-z*g^2)/((1-z-z*g)*(1+z-3*z*g)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
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CROSSREFS
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Cf. A128724.
Sequence in context: A037696 A026895 A037605 this_sequence A055146 A014827 A141845
Adjacent sequences: A128723 A128724 A128725 this_sequence A128727 A128728 A128729
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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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