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A128734
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Number of LD's in all skew Dyck paths of semilength n.
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2
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0, 0, 0, 1, 8, 48, 261, 1358, 6907, 34684, 172850, 857389, 4240442, 20933422, 103221134, 508623877, 2505298946, 12338127048, 60761615904, 299256606347, 1474086307696, 7262524940428, 35789196572489, 176410731649052
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OFFSET
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0,5
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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..floor((n-1)/2)} k*A128733(n,k), for n >= 1.
G.f.: z^2*g^2*(g-1)/(1-3*z+2*z^2-3*z^2*g^2), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6*z + 5*z^2))/(2*z).
Conjecture: +2*(3*n-20)*(n-3)*(n+1)*a(n) +(-39*n^3+362*n^2-739*n+300)*a(n-1) +8*(n-2) *(6*n^2-49*n+75) *a(n-2) -5*(n-2)*(n-3)*(3*n-17)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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a(4)=8 because we have UDUUUD(LD), UUUD(LD)UD, UUDUUD(LD), UUUUD(LD)D, UUUDUD(LD), UUUUDD(LD), UUUUDL(LD) and UUUUD(LD)L (the LD's are shown between parentheses; the other 28 skew Dyck paths of semilength 4 contain no LD).
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=z^2*g^2*(g-1)/(1-3*z+2*z^2-3*z^2*g^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[x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-Sqrt[1-6*x+5*x^2])/2/x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) x='x+O('x^50); concat([0, 0, 0], Vec(x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2*((1-x-sqrt(1-6*x+5*x^2))/2/x-1)/(1-3*x+2*x^2-3*x^2*((1-x-sqrt(1-6*x+5*x^2))/2/x)^2))) \\ G. C. Greubel, Mar 20 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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