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A128743
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Number of UU's (i.e. doublerises) 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 a path is defined to be the number of its steps.
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1
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0, 0, 2, 13, 69, 346, 1700, 8286, 40264, 195488, 949302, 4613025, 22436997, 109240038, 532410060, 2597468685, 12684628125, 62002335160, 303332650190, 1485213237135, 7277719953415, 35687662907750, 175120787451540
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OFFSET
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0,3
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COMMENTS
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a(n)=Sum[k*A128718(n,k), k=0..n-1].
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LINKS
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Table of n, a(n) for n=0..22.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
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G.f.=[1-4z+z^2+(z-1)sqrt(1-6z+5z^2)]/[2z*sqrt(1-6z+5z^2)].
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EXAMPLE
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a(2)=2 because the paths of semilength 2 are UDUD, UUDD and UUDL, having altogether 2 UU's.
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MAPLE
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G:=(1-4*z+z^2+(z-1)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..25);
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CROSSREFS
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Cf. A128718.
Sequence in context: A038144 A097977 A136780 * A218184 A188676 A097349
Adjacent sequences: A128740 A128741 A128742 * A128744 A128745 A128746
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Mar 30 2007
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STATUS
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approved
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