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Search: id:A129171
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| A129171 |
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Sum of the heights of the peaks 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, 1, 6, 32, 165, 840, 4251, 21443, 107946, 542680, 2725635, 13679997, 68623176, 344090307, 1724754180, 8642952000, 43300971885, 216895107480, 1086253033035, 5439405705125, 27234492215400, 136345625309965, 682531666024170
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n)=Sum(k*A129170(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[3-3z-sqrt(1-6z+5z^2)]/(1-6z+5z^2).
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EXAMPLE
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a(2)=6 because in the 3 skew Dyck paths of semilength 2, namely UDUD, UUDD and UUDL, the heights of the peaks are 1,1,2 and 2.
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MAPLE
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G:=z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
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CROSSREFS
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Cf. A129170.
Sequence in context: A097139 A034942 A046714 this_sequence A082585 A084326 A137637
Adjacent sequences: A129168 A129169 A129170 this_sequence A129172 A129173 A129174
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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), Apr 07 2007
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