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Search: id:A128737
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| A128737 |
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Number of LDU'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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2
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| 0, 0, 0, 0, 1, 10, 69, 412, 2291, 12244, 63886, 328256, 1669363, 8429384, 42349096, 211982828, 1058244079, 5272285552, 26227527576, 130323237088, 647013004499, 3210128312122, 15919166804461, 78915323039268, 391100149306301
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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a(n)=Sum(k*A128735(n,k), k>=0).
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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(g-1)^3/(4g-2zg-6zg^2-3+3*z), 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)=1 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series(z*(g-1)^3/(4*g-2*z*g-6*z*g^2-3+3*z), z=0, 30): seq(coeff(ser, z, n), n=0..27);
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CROSSREFS
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Cf. A128735.
Sequence in context: A081280 A038806 A016273 this_sequence A130548 A090084 A025221
Adjacent sequences: A128734 A128735 A128736 this_sequence A128738 A128739 A128740
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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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