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A128721
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Number of UUU's in all skew Dyck paths of semilength n.
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2
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0, 0, 0, 4, 28, 157, 820, 4155, 20742, 102725, 506504, 2491230, 12236520, 60063399, 294748884, 1446436680, 7099442700, 34855583275, 171187439920, 841084246980, 4134129246180, 20328683526575, 100003531112300, 492153054177155
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OFFSET
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0,4
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COMMENTS
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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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LINKS
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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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a(n) = Sum_{k=0..n-2} k*A128719(n,k) (n >= 2).
G.f.: (2zg - g + 1 - z + z^2)/(2zg + z - 1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Recurrence: 2*(n+1)*(121*n-348)*a(n) = (1663*n^2 - 4620*n + 1392)*a(n-1) - (2476*n^2 - 11133*n + 11787)*a(n-2) + 5*(n-4)*(211*n-537)*a(n-3). - Vaclav Kotesovec, Nov 19 2012
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EXAMPLE
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a(3)=4 because each of the paths UUUDDD, UUUDLD, UUUDDL and UUUDLL contains one UUU, while the other six paths of semilength 3 contain no UUU's.
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MAPLE
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G:=(1-5*z+4*z^2-2*z^3-(1-2*z)*sqrt(1-6*z+5*z^2))/2/z/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 28): seq(coeff(Gser, z, n), n=0..25);
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MATHEMATICA
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CoefficientList[Series[(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)-(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+1-x+x^2)/(2*x*(1-x-Sqrt[1-6*x+5*x^2])/(2*x)+x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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