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A128748
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Number of peaks at height >1 in all skew Dyck paths of semilength n.
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2
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0, 2, 11, 54, 260, 1247, 5982, 28741, 138364, 667488, 3226503, 15625476, 75802578, 368316888, 1792203759, 8732274312, 42598366616, 208036945958, 1017023261529, 4976560342522, 24372741339016, 119461561111023, 585970198529224
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OFFSET
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1,2
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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..n-1} A128747(n,k).
G.f.: (1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-z)*sqrt(1-6*z+5*z^2)).
D-finite with recurrence 2*(n+2)*a(n) +(-19*n-18)*a(n-1) +(53*n-12)*a(n-2) +2*(-20*n+19)*a(n-3) +(-n+26)*a(n-4) +5*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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a(2)=2 because in the paths UDUD, U(UD)D and U(UD)L we have altogether 2 peaks at height >1 (shown between parentheses).
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MAPLE
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G:=(1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/2/z/(2-z)/sqrt(1-6*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);
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MATHEMATICA
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Rest[CoefficientList[Series[(1-4*x+2*x^2+x^3-(1-x+x^2)*Sqrt[1-6*x+5*x^2]) /2/x/(2-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], Vec((1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-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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