cp's OEIS Frontend

A190168 Number of peakless Motzkin paths of length n having no (1,0)-steps at levels 1,3,5,... .

%I A190168 #11 Jul 26 2022 13:14:13
%S A190168 1,1,1,1,1,2,4,7,12,21,38,70,130,243,457,865,1647,3152,6059,11693,
%T A190168 22647,44007,85770,167626,328430,644993,1269413,2503339,4945897,
%U A190168 9788700,19404866,38526335,76599502,152503123,304006284,606745700,1212335896,2424964327,4855454654
%N A190168 Number of peakless Motzkin paths of length n having no (1,0)-steps at levels 1,3,5,... .
%C A190168 a(n) = A190167(n,0).
%H A190168 Matthew House, <a href="/A190168/b190168.txt">Table of n, a(n) for n = 0..3142</a>
%F A190168 G.f. G=G(z) satisfies the equation z^2*(1-z+z^2)G^2-(1+z^2)(1-z+z^2)G +1+z^2=0.
%F A190168 G.f.: (1+1/x^2-sqrt(1+1/x^4-2/x^2-4/x-(4-4*x)/(1-x+x^2)))/2. - _Matthew House_, Feb 12 2017
%F A190168 D-finite with recurrence (n+2)*a(n) +2*(-n-1)*a(n-1) +(n+2)*a(n-2) +2*(-n+2)*a(n-3) +2*(-n+4)*a(n-5) +(n-8)*a(n-6) +2*(-n+7)*a(n-7) +(n-8)*a(n-8)=0. - _R. J. Mathar_, Jul 26 2022
%e A190168 a(5)=2 because we have hhhhh and uuhdd, where u=(1,1), h=(1,0), d=(1,-1).
%p A190168 eq := z^2*(1-z+z^2)*G^2-(1+z^2)*(1-z+z^2)*G+1+z^2=0: g:=RootOf(eq,G): Gser:=series(g,z=0,46): seq(coeff(Gser,z,n),n=0..38);
%t A190168 CoefficientList[Series[(1 + 1/x^2 - Sqrt[1 + 1/x^4 - 2/x^2 - 4/x - (4 - 4 x)/(1 - x + x^2)])/2, {x, 0, 38}], x] (* _Michael De Vlieger_, Feb 12 2017 *)
%Y A190168 Cf. A190167, A190165
%K A190168 nonn
%O A190168 0,6
%A A190168 _Emeric Deutsch_, May 06 2011