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A105641
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Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1).
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1
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0, 1, 2, 5, 14, 39, 111, 322, 947, 2818, 8470, 25677, 78420, 241061, 745265, 2315794, 7228702, 22656505, 71273364, 224965675, 712249471, 2261326010, 7197988973, 22966210236, 73437955105, 235307698544, 755395560220, 2429293941019
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OFFSET
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2,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: [(1+z)^2-sqrt((1+z^2)^2-4z)]/[2z(2+z+z^2)]-1.
D-finite with recurrence 2*(n+1)*a(n) +(-7*n+5)*a(n-1) +(n-5)*a(n-2) +2*(-n-1)*a(n-3) +2*(2*n-7)*a(n-4) +(n-5)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
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EXAMPLE
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a(4)=2 because we have UUDUDUDD and UUUDUDDD.
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MAPLE
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G:=((1+z)^2-sqrt((1+z^2)^2-4*z))/2/z/(2+z+z^2)-1: Gser:=series(G, z=0, 36): seq(coeff(Gser, z^n), n=2..32);
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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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