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

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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)

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 *)

Prog

(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

Crossrefs

Cf. A001006, A129400.

Sequence in context: A081341 A355353 A363439 * A050623 A037760 A037648

Adjacent sequences: A132897 A132898 A132899 * A132901 A132902 A132903

Keyword

easy,nonn

Author

Paul Barry, Sep 04 2007

Status

approved