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A129169
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Number of UU's starting at level 0 in all skew Dyck paths of semilength n.
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2
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0, 0, 2, 9, 37, 153, 643, 2745, 11881, 52038, 230268, 1028007, 4625013, 20949444, 95462022, 437318631, 2012933893, 9304997544, 43179620542, 201076887771, 939358727203, 4401177906249, 20676208303727, 97374640613139
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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, 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,..,floor(n/2)} k*A129168(n,k).
G.f.: 2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^2.
Recurrence: 2*n*(n+2)*a(n) = (13*n^2+8*n+4)*a(n-1) - (16*n^2-7*n-18)*a(n-2) + 5*(n-2)*(n+1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
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EXAMPLE
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a(3)=9 because in the 10 skew Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, UD(UU)DL, (UU)DDUD, (UU)DUDD, (UU)UDDD, (UU)UDLD, (UU)DUDL, (UU)UDDL and (UU)UDLL, we have altogether 9 UU's starting at level 0 (shown between parentheses).
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MAPLE
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G:=2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
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MATHEMATICA
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CoefficientList[Series[2*(2-x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(1+x+Sqrt[1-6*x+5*x^2])^2, {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, 0], Vec(2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^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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STATUS
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approved
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