cp's OEIS Frontend

A190171 Number of peakless Motzkin paths of length n having no UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).

%I A190171 #10 Jul 22 2022 11:56:55
%S A190171 1,1,1,1,2,5,12,27,60,135,309,717,1680,3966,9423,22518,54091,130540,
%T A190171 316358,769577,1878497,4599623,11294640,27807381,68627188,169746823,
%U A190171 420732391,1044830875,2599352149,6477571270,16167429874,40411920571,101153167258,253522241008
%N A190171 Number of peakless Motzkin paths of length n having no UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).
%C A190171 a(n)=A190170(n,0).
%F A190171 G.f. G=G(z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z) and S=1+zS+z^2*S(S-1).
%F A190171 From _Vaclav Kotesovec_, May 29 2022: (Start)
%F A190171 G.f.: 2/(1 + sqrt((1 + (-3 + x)*x)*(1 + x + x^2)) + x*(-1 + x + 2*x^2)).
%F A190171 a(n) ~ 5^(1/4) * phi^(2*n+6) / (18 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)
%F A190171 Conjecture D-finite with recurrence (n+2)*a(n) +(-n+1)*a(n-1) +2*(-2*n-1)*a(n-2) +9*a(n-3) +(-n+1)*a(n-4) -9*a(n-5) +2*(-2*n+5)*a(n-6) +(-n+1)*a(n-7) +(n-4)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022
%e A190171 a(4)=2 because we have HHHH and UHHD.
%p A190171 p1 := G-1-z*G-z^2*G*(S-1-z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 40)): seq(coeff(Gser, z, n), n = 0 .. 33);
%t A190171 CoefficientList[Series[2/(1 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)] + x*(-1 + x + 2*x^2)), {x, 0, 40}], x] (* _Vaclav Kotesovec_, May 29 2022 *)
%Y A190171 Cf. A190170.
%K A190171 nonn
%O A190171 0,5
%A A190171 _Emeric Deutsch_, May 06 2011