A190165 Number of peakless Motzkin paths of length n having no (1,0)-steps at levels 0,2,4,... .
1, 0, 0, 1, 1, 1, 2, 4, 7, 12, 22, 41, 76, 142, 268, 509, 971, 1861, 3583, 6925, 13430, 26128, 50980, 99735, 195594, 384454, 757256, 1494465, 2954715, 5851677, 11607348, 23058492, 45870685, 91371464, 182231978, 363871075, 727364502, 1455503056, 2915461721, 5845386764, 11730347948
Offset: 0
Keywords
Examples
a(6)=2 because we have uhduhd and uhhhhd, where u=(1,1), h=(1,0), d=(1,-1).
Programs
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Maple
eq := z^2*(1+z^2)*G^2-(1+z^2)*(1-z+z^2)*G+1-z+z^2 =0: g:=RootOf(eq,G): Gser:=series(g,z=0,46): seq(coeff(Gser,z,n),n=0..40);
Formula
G.f. G=G(z) satisfies the equation z^2*(1+z^2)G^2 - (1+z^2)(1-z+z^2)G + 1-z+z^2=0.
D-finite with recurrence +(n+2)*a(n) +(-2*n-1)*a(n-1) +(n+1)*a(n-2) +(-2*n+1)*a(n-3) +(-2*n+11)*a(n-5) +(n-7)*a(n-6) +(-2*n+13)*a(n-7) +(n-8)*a(n-8)=0. - R. J. Mathar, Jul 24 2022
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