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A132900
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Colored Motzkin paths where each of the steps has three possible colors.
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1
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1, 3, 18, 108, 729, 5103, 37179, 277749, 2119203, 16435305, 129199212, 1027098306, 8243181351, 66698502705, 543507899346, 4456368744804, 36738955831707, 304354824214977, 2532328310730798, 21152326520189628, 177310026608555619, 1491097815365481477
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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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) = 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
G.f.: 1/G(x), with G(x) = 1-3*x-9*x^2/G(x) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 01 2023
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)
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MAPLE
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seq(9^n * simplify(hypergeom([3/2, -n], [3], 4/3)), n = 0..20); # Peter Bala, Feb 04 2024
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MATHEMATICA
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CoefficientList[Series[(1-3*x-Sqrt[1-6*x-27*x^2])/(18*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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