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A129160
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Sum of the semi-abscissae of the first returns to the axis over all skew Dyck paths of semilength n.
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2
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1, 4, 18, 82, 378, 1760, 8262, 39044, 185526, 885596, 4243590, 20400954, 98353278, 475322352, 2302064010, 11170370850, 54293503770, 264290420540, 1288257980310, 6287181414470, 30717958762350, 150234512678480, 735446569221810, 3603330368706640, 17668505697688098, 86698739895529300
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OFFSET
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1,2
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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 the 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=1,..,n} k*A129159(n,k).
G.f.: x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2).
Recurrence: n*(3*n-1)*a(n) = 18*(n-1)*n*a(n-1) - 5*(n-3)*(3*n+2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
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EXAMPLE
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a(2)=4 because UDUD, UUDD and UUDL yield 1+2+1=4.
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MAPLE
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G:=z-1+(1-3*z+2*z^2)/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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MATHEMATICA
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CoefficientList[Series[(1/x) (x - 1 + (1 - 3*x + 2*x^2)/Sqrt[1 - 6*x + 5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(PARI) x='x+O('x^25); Vec(x-1+(1-3*x+2*x^2)/sqrt(1-6*x+5*x^2)) \\ G. C. Greubel, Feb 09 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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