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A132900 Colored Motzkin paths where each of the steps has three possible colors. 1
1, 3, 18, 108, 729, 5103, 37179, 277749, 2119203, 16435305, 129199212, 1027098306, 8243181351, 66698502705, 543507899346, 4456368744804, 36738955831707, 304354824214977, 2532328310730798, 21152326520189628 (list; graph; refs; listen; history; text; internal format)
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

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) 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. A129400.

Sequence in context: A267662 A169604 A081341 * A050623 A037760 A037648

Adjacent sequences:  A132897 A132898 A132899 * A132901 A132902 A132903

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 04 2007

STATUS

approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)