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A128096
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Number of steps that touch the x-axis in all peakless Motzkin paths of length n.
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1
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1, 2, 5, 12, 27, 62, 144, 336, 790, 1870, 4452, 10656, 25629, 61910, 150145, 365450, 892434, 2185928, 5369097, 13221422, 32634935, 80730942, 200116410, 496992992, 1236482727, 3081389406, 7690966549, 19224282880, 48119034729, 120599916654
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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-z^2-sqrt((1+z+z^2)(1-3z+z^2))]/[1-z+z^2+sqrt((1+z+z^2)(1-3z+z^2))]^2.
Conjecture: -2*(n+4)*(1088*n-4241)*a(n) +(6616*n^2-12361*n-59102)*a(n-1) +2*(-1176*n^2+6520*n-4985)*a(n-2) +(2088*n^2-6071*n+16522) *a(n-3) +2*(-3352*n^2+22882*n-35653)*a(n-4) +(2264*n-6277)*(n-6) *a(n-5)=0. - R. J. Mathar, Jun 17 2016
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EXAMPLE
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a(3)=5 because in the peakless Motzkin paths of length 3 (namely HHH 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-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(1-z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2)))^2: gser:=series(g, z=0, 38): seq(coeff(gser, z, n), n=1..35);
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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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