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 A025226 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also a(n) = 3^n*C(n-1), where C = A000108 (Catalan numbers). 6
 3, 9, 54, 405, 3402, 30618, 288684, 2814669, 28146690, 287096238, 2975361012, 31241290626, 331638315876, 3553267670100, 38375290837080, 417331287853245, 4566095267100210 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Total number of rows in all Kleene truth tables for bracketed implication with n distinct variables. See Yildiz link. - Michel Marcus, Oct 21 2020 LINKS G. C. Greubel, Table of n, a(n) for n = 1..900 Volkan Yildiz, Counting with 3-valued truth tables of bracketed formulae connected by implication, arXiv:2010.10303 [math.GM], 2020. Volkan Yildiz, Notes on algebraic structure of truth tables of bracketed formulae connected by implications, arXiv:2106.04728 [math.CO], 2021. FORMULA a(n) = Sum_{j=1..n-1} a(j)*a(n-j), with a(1) = 3. a(n) = 3^n * A000108(n-1). G.f.: (1-sqrt(1-12*x))/2. - Michael Somos, Jun 08 2000 Given g.f. C(x) and given A(x)= g.f. of A100239, then B(x) = A(x) - (1+2*x) satisfies B(x) = x - C(x*B(x)). - Michael Somos, Sep 07 2005 G.f.: (1 - U(0))/x where U(k)= 1 - 3*x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 30 2012 D-finite with recurrence: n*a(n) +6*(3-2*n)*a(n-1) = 0. - R. J. Mathar, Nov 12 2012 a(n) = 3^n/(4*n-2)*binomial(2*n,n). - Vaclav Kotesovec, Oct 11 2013 EXAMPLE a(3) = 3^3*C(2) = 27*2 = 54. MATHEMATICA Rest[CoefficientList[Series[(1-Sqrt[1-12x])/2, {x, 0, 20}], x]] (* Harvey P. Dale, Mar 09 2011 *) PROG (PARI) a(n)=polcoeff((1-sqrt(1-12*x+x*O(x^n)))/2, n) (Magma) [3^n*Catalan(n-1): n in [1..30]]; // G. C. Greubel, May 20 2022 (SageMath) [3^n*catalan_number(n-1) for n in (1..30)] # G. C. Greubel, May 20 2022 CROSSREFS Cf. A000108, A005159. Sequence in context: A363442 A212418 A337039 * A001194 A032179 A233189 Adjacent sequences: A025223 A025224 A025225 * A025227 A025228 A025229 KEYWORD nonn AUTHOR Clark Kimberling STATUS approved

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Last modified September 12 09:47 EDT 2024. Contains 375850 sequences. (Running on oeis4.)