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A129166
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Number of skew Dyck paths of semilength n with no base pyramids.
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2
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1, 0, 1, 5, 19, 73, 292, 1203, 5065, 21697, 94274, 414514, 1840981, 8247011, 37220261, 169079113, 772489020, 3547371679, 16364309243, 75799327800, 352402156770, 1643878188646, 7691841654538, 36091803172733
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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 the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
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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.: (1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2))).
D-finite with recurrence 6*(n+1)*a(n) +2*(-25*n+11)*a(n-1) +(131*n-229)*a(n-2) +2*(-92*n+261)*a(n-3) +2*(81*n-311)*a(n-4) +(-91*n+439)*a(n-5) +(31*n-183)*a(n-6) +5*(-n+7)*a(n-7)=0. - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(2)=1 because we have UUDL.
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MAPLE
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G:=(1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^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[(1-x)*(3-3*x-Sqrt[1-6*x+5*x^2])/(2-(1-x)*(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); Vec((1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^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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