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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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1
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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; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| a(n)=Sum(A128738(n,k), k>=0).
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| 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
| 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
| 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
| Cf. A128738.
Sequence in context: A094951 A099621 A056015 * A026705 A003463 A026771
Adjacent sequences: A128737 A128738 A128739 * A128741 A128742 A128743
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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