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A128730
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Number of UDL's in all skew Dyck paths of semilength n.
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3
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0, 0, 1, 4, 16, 68, 301, 1366, 6301, 29400, 138355, 655424, 3121438, 14930540, 71675839, 345148892, 1666432816, 8064278288, 39103576699, 189949958332, 924163714216, 4502711570988, 21966152501239, 107284324830302
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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 steps in it.
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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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G.f.: 2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)).
D-finite with recurrence: +2*(n-1)*(3*n-8)*a(n) +(-39*n^2+161*n-148)*a(n-1) +(48*n^2-215*n+220)*a(n-2) -5*(3*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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a(3) = 4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL (the other six skew Dyck paths of semilength are the five Dyck paths and UUUDDL.
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MAPLE
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G:=2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..26);
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MATHEMATICA
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CoefficientList[Series[2*x^2/(1-6*x+5*x^2+(1+x)*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) z='z+O('z^50); concat([0, 0], Vec(2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 19 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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