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Search: id:A128740
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| A128740 |
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Number of DD'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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+0
2
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| 0, 0, 1, 6, 31, 154, 754, 3670, 17824, 86524, 420169, 2041946, 9932959, 48368000, 235769011, 1150413818, 5618786629, 27468246832, 134399280931, 658139933938, 3225323325109, 15817633139722, 77625378841756, 381190465089138
(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(A128738(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.=(2zg-g-z+1)/(3zg-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)=6 because each of the paths UDUUDD, UUDDUD, UUDUDD, UUUDDL contains one DD, the path UUUDDD contains 2 DD's, and the paths UDUDUD, UDUUDL, UUUDLD, UUDUDL, and UUUDLL contain no DD's.
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series((2*z*g-g-z+1)/(3*z*g-z-1), z=0, 30): seq(coeff(ser, z, n), n=0..27);
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CROSSREFS
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Cf. A128738.
Adjacent sequences: A128737 A128738 A128739 this_sequence A128741 A128742 A128743
Sequence in context: A094951 A099621 A056015 this_sequence A026705 A003463 A026771
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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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