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%I #6 Jul 24 2022 13:35:18
%S 1,0,0,1,1,1,2,4,7,12,22,41,76,142,268,509,971,1861,3583,6925,13430,
%T 26128,50980,99735,195594,384454,757256,1494465,2954715,5851677,
%U 11607348,23058492,45870685,91371464,182231978,363871075,727364502,1455503056,2915461721,5845386764,11730347948
%N Number of peakless Motzkin paths of length n having no (1,0)-steps at levels 0,2,4,... .
%C a(n) = A190164(n,0).
%F 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.
%F 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
%e a(6)=2 because we have uhduhd and uhhhhd, where u=(1,1), h=(1,0), d=(1,-1).
%p 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);
%Y Cf. A190164, A190167, A190168
%K nonn
%O 0,7
%A _Emeric Deutsch_, May 06 2011
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