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%I #5 Jun 17 2016 13:42:42
%S 1,2,5,12,27,62,144,336,790,1870,4452,10656,25629,61910,150145,365450,
%T 892434,2185928,5369097,13221422,32634935,80730942,200116410,
%U 496992992,1236482727,3081389406,7690966549,19224282880,48119034729,120599916654
%N Number of steps that touch the x-axis in all peakless Motzkin paths of length n.
%C a(n)=Sum(k*A128095(n,k), k=1..n).
%F 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.
%F 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
%e 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.
%p 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);
%Y Cf. A128095.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Feb 14 2007
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