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A107264 Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2). 7
1, 3, 12, 54, 261, 1323, 6939, 37341, 205011, 1143801, 6466230, 36960300, 213243435, 1240219269, 7263473148, 42799541886, 253556163243, 1509356586897, 9023497273548, 54154973176074, 326154592965879, 1970575690572297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Series reversion of x/(1+3x+3x^2). Transform of 3^n under the matrix A107131. A row of A107267.

Counts colored Motzkin paths, where H(1,0) and U(1,1) each have 3 colors and D(1,-1) one color. - Paul Barry, May 18 2005

Number of Motzkin paths of length n in which both the "up" and the "level" steps come in three colors. - Paul Barry, May 18 2005

Third binomial transform of 1,0,3,0,18,0,... or 3^n*C(n) (A005159) with interpolated zeros. - Paul Barry, May 24 2005

As a continued fraction, the g.f. is 1/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(1-3*x-3*x^2/(.... - Paul Barry, Dec 02 2008

LINKS

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

N. Gabriel, K. Peske, L. Pudwell, S. Tay, Pattern Avoidance in Ternary Trees, J. Int. Seq. 15 (2012) # 12.1.5.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

L. Pudwell, Pattern avoidance in trees, (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013

FORMULA

G.f.: (1 - 3x - sqrt(1-6x-3x^2))/(6x^2);

a(n) = Sum_{k=0..n} (1/(k+1))*C(k+1, n-k+1)*C(n, k)3^k.

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*C(k)*3^(n-k). - Paul Barry, May 18 2005

E.g.f.: exp(3x)*Bessel_I(1, sqrt(3)*2*x)/(sqrt(3)*x). - Paul Barry, May 24 2005

a(n) = (1/Pi)*Integral_{x=3-2*sqrt(3)..3+2*sqrt(3)} x^n*sqrt(-x^2 + 6*x + 3)/6. - Paul Barry, Sep 16 2006

a(n) = A156016(n+1)/3. - Philippe Deléham, Feb 04 2009

Recurrence: (n+2)*a(n) = 3*(2*n+1)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ (5+3*sqrt(3))*(3+2*sqrt(3))^n/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

MATHEMATICA

CoefficientList[Series[(1-3*x-Sqrt[1-6*x-3*x^2])/(6*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

CROSSREFS

Sequence in context: A054666 A006026 A158826 * A200740 A177133 A186241

Adjacent sequences:  A107261 A107262 A107263 * A107265 A107266 A107267

KEYWORD

easy,nonn

AUTHOR

Paul Barry, May 15 2005

STATUS

approved

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Last modified December 12 07:31 EST 2019. Contains 329948 sequences. (Running on oeis4.)