A132900 Colored Motzkin paths where each of the steps has three possible colors.
1, 3, 18, 108, 729, 5103, 37179, 277749, 2119203, 16435305, 129199212, 1027098306, 8243181351, 66698502705, 543507899346, 4456368744804, 36738955831707, 304354824214977, 2532328310730798, 21152326520189628, 177310026608555619, 1491097815365481477
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
-
Maple
seq(9^n * simplify(hypergeom([3/2, -n], [3], 4/3)), n = 0..20); # Peter Bala, Feb 04 2024
-
Mathematica
CoefficientList[Series[(1-3*x-Sqrt[1-6*x-27*x^2])/(18*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *) -
PARI
my(x='x+O('x^50)); Vec((1-3*x-sqrt(1-6*x-27*x^2))/(18*x^2)) \\ G. C. Greubel, Mar 21 2017
Formula
G.f.: (1-3*x-sqrt(1-6*x-27*x^2))/(18*x^2).
G.f. is the reversion of x/(1+3*x+9*x^2).
a(n) = 3^n * A001006(n).
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(k)*3^(n-2k)*3^k*3^k, where C(n) = A000108(n).
a(n) = (1/(2*Pi))*Integral_{x=-3..9} x^n*sqrt(27 + 6x - x^2)/9.
Conjecture: (n+2)*a(n) - 3*(2*n+1)*a(n-1) + 27*(1-n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 3^(2*n+3/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
G.f.: 1/G(x), with G(x) = 1-3*x-9*x^2/G(x) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 01 2023
From Peter Bala, Feb 02 2024: (Start)
G.f.: 1/(1 + 3*x)*c(3*x/(1 + 3*x))^2 = 1/(1 - 9*x)*c(-3*x/(1 - 9*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers.
a(n) = 3^n *Sum_{k = 0..n} (-1)^(n+k)*binomial(n,k)*Catalan(k+1).
a(n) = 9^n * Sum_{k = 0..n} (-3)^(-k)*binomial(n,k)*Catalan(k+1). (End)