A107264 Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2).

1, 3, 12, 54, 261, 1323, 6939, 37341, 205011, 1143801, 6466230, 36960300, 213243435, 1240219269, 7263473148, 42799541886, 253556163243, 1509356586897, 9023497273548, 54154973176074, 326154592965879, 1970575690572297

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

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

N. Gabriel, K. Peske, L. Pudwell, and 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

D-finite with 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

G.f.: Let F(x) be the g.f. of A348189 with offset 1, then F(x) = x + 2*x^2*F(x)^2*A(x*F(x)). - Alexander Burstein, Feb 14 2022

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 * A370441 A200740 A177133

Adjacent sequences: A107261 A107262 A107263 * A107265 A107266 A107267

Keyword

easy,nonn

Author

Paul Barry, May 15 2005

Status

approved