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URL: https://oeis.org/A128096

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A128096
Number of steps that touch the x-axis in all peakless Motzkin paths of length n.
1
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
OFFSET
1,2
COMMENTS
a(n)=Sum(k*A128095(n,k), k=1..n).
FORMULA
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
EXAMPLE
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.
MAPLE
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);
CROSSREFS
Cf. A128095.
Sequence in context: A077863 A319172 A018009 * A018010 A303022 A026710
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 14 2007
STATUS
approved