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 A107264 Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2). 9
 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, 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 * 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 July 5 20:09 EDT 2022. Contains 355102 sequences. (Running on oeis4.)