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A128098
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Number of steps that touch the x-axis in all Motzkin paths of length n.
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1
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1, 4, 11, 30, 80, 214, 574, 1548, 4197, 11440, 31339, 86252, 238407, 661584, 1842585, 5148960, 14432643, 40569804, 114339777, 323031750, 914683602, 2595411126, 7378861196, 21016701652, 59962687675, 171353419536, 490407962229
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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G.f.=4[1-sqrt(1-2z-3z^2)]/[1-z+sqrt(1-2z-3z^2)]^2.
D-finite with recurrence (n-1)*(16*n-27)*(n+4)*a(n) -2*n*(16*n^2-3*n-28)*a(n-1) -3*n*(16*n-11)*(n-1)*a(n-2)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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a(3)=11 because in the Motzkin paths of length 3 (namely HHH, HUD, UDH and UHD, where H=(1,0), U=(1,1) and D=(1,-1)) all the steps, with the exception of H in UHD, touch the x-axis.
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MAPLE
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g:=4*(1-sqrt(1-2*z-3*z^2))/(1-z+sqrt(1-2*z-3*z^2))^2: gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=1..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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