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A129162
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Sum of heights of 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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1, 5, 23, 103, 462, 2086, 9493, 43521, 200855, 932429, 4350995, 20395349, 95987113, 453354623, 2148027772, 10206485598, 48621125308, 232156538970, 1110842790406, 5325499426116, 25576096186920, 123030491611330
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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
| a(n)=Sum(k*A129161(n,k), k=1..n).
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EXAMPLE
| a(2)=5 because the paths UDUD, UUDD and UUDL have heights 1, 2 and 2, respectively.
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MAPLE
| H[0]:=1: for k from 1 to 32 do H[k]:=simplify((1+z*H[k-1]-z)/(1-z*H[k-1])) od: for k from 1 to 32 do h[k]:=factor(simplify(H[k]-H[k-1])) od: for k from 1 to 32 do hser[k]:=series(h[k], z=0, 30) od: T:=(n, k)->coeff(hser[k], z, n): seq(add(k*T(n, k), k=1..n), n=1..25);
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CROSSREFS
| Cf. A129161.
Sequence in context: A120902 A054441 A102285 * A167660 A026760 A064914
Adjacent sequences: A129159 A129160 A129161 * A129163 A129164 A129165
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 03 2007
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