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.

1, 3, 6, 13, 30, 71, 171, 417, 1026, 2542, 6333, 15849, 39813, 100329, 253518, 642117, 1629726, 4143857, 10553511, 26916426, 68739015, 175752268, 449846001, 1152528593, 2955487605, 7585165701, 19481930556, 50073211027, 128784497466, 331426205715, 853409723277

Offset

0,2

Comments

This generating function, together with the multiplier function -xg(x), produce an involution in the Riordan group.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

Formula

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).

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

a(n) ~ 5^(1/4) * phi^(2*n+1) / sqrt(Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A004148.

Sequence in context: A098075 A137584 A201631 * A141353 A130582 A126296

Adjacent sequences: A125264 A125265 A125266 * A125268 A125269 A125270

Keyword

nonn

Author

Louis Shapiro and Gi-Sang Cheon, Jan 15 2007

Status

approved