login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A107264 Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2). 13
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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 15 2005
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)