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Search: id:A128748
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| A128748 |
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Number of peaks at height >1 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, 2, 11, 54, 260, 1247, 5982, 28741, 138364, 667488, 3226503, 15625476, 75802578, 368316888, 1792203759, 8732274312, 42598366616, 208036945958, 1017023261529, 4976560342522, 24372741339016, 119461561111023, 585970198529224
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=Sum(A128747(n,k), k=0..n-1).
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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.=[1-4z+2z^2+z^3-(1-z+z^2)sqrt(1-6z+5z^2)]/[2z(2-z)sqrt(1-6z+5z^2)].
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EXAMPLE
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a(2)=2 because in the paths UDUD, U(UD)D, and U(UD)L we have altogether 2 peaks at height >1 (shown between parentheses).
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MAPLE
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G:=(1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/2/z/(2-z)/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);
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CROSSREFS
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Cf. A128747.
Sequence in context: A052171 A030281 A063767 this_sequence A037522 A037731 A115205
Adjacent sequences: A128745 A128746 A128747 this_sequence A128749 A128750 A128751
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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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