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

Author

Paul Barry, May 15 2005

Keywords

• easy
• nonn

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

Programs

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 *)

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