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A129173
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Total area below all skew Dyck paths of semilength n.
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2
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0, 1, 9, 58, 336, 1853, 9945, 52487, 273939, 1418567, 7303791, 37441560, 191287254, 974642943, 4955123955, 25146686730, 127424717400, 644873878895, 3260055588615, 16465301636090, 83092583965020, 419031686115875
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OFFSET
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0,3
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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 and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.
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FORMULA
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a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).
G.f.: (1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)).
(n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.
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EXAMPLE
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a(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.
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MAPLE
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a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n], n=0..25);
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MATHEMATICA
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CoefficientList[Series[(1+x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*(1-5*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(PARI) z='z +O('z^25); concat([0], Vec((1+z)*(1-3*z-sqrt(1-6*z+5*z^2))/(2*z*(1-5*z)))) \\ G. C. Greubel, Feb 10 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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STATUS
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approved
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