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A125267 Number of Motzkin paths with no peaks and with level steps at height 0 having three colors except that consecutive level steps at height 0 must have different colors.

%I A125267 #19 May 29 2022 04:18:33
%S A125267 1,3,6,13,30,71,171,417,1026,2542,6333,15849,39813,100329,253518,
%T A125267 642117,1629726,4143857,10553511,26916426,68739015,175752268,
%U A125267 449846001,1152528593,2955487605,7585165701,19481930556,50073211027,128784497466,331426205715,853409723277
%N A125267 Number of Motzkin paths with no peaks and with level steps at height 0 having three colors except that consecutive level steps at height 0 must have different colors.
%C A125267 This generating function, together with the multiplier function -xg(x), produce an involution in the Riordan group.
%H A125267 Alois P. Heinz, <a href="/A125267/b125267.txt">Table of n, a(n) for n = 0..1000</a>
%H A125267 Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%F A125267 G.f.: (g(x)*(1+x))/(1-x*g(x)) where g(x)=((1-x+x^2)-sqrt((1-x+x^2)^2-4x^2))/(2*x^2).
%F A125267 Conjecture: -(n+1)*(n-2)*a(n) +2*(n^2-n-3)*a(n-1) +(n^2-3*n+8)*a(n-2) +2*(n^2-5*n+3)*a(n-3) -(n-1)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 17 2016
%F A125267 a(n) ~ 5^(1/4) * phi^(2*n+1) / sqrt(Pi*n), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, May 29 2022
%e A125267 a(3) = 13 since there are 12 = 3*2*2 paths that stay at level 0 and one path ULD that goes above level 0.
%t A125267 CoefficientList[Series[(((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)*(1 + x))/(1 - x*((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jan 10 2017 *)
%Y A125267 Cf. A004148.
%K A125267 nonn
%O A125267 0,2
%A A125267 _Louis Shapiro_ and Gi-Sang Cheon, Jan 15 2007