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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).
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%I #37 May 21 2022 08:29:30

%S 3,9,54,405,3402,30618,288684,2814669,28146690,287096238,2975361012,

%T 31241290626,331638315876,3553267670100,38375290837080,

%U 417331287853245,4566095267100210

%N 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).

%C Total number of rows in all Kleene truth tables for bracketed implication with n distinct variables. See Yildiz link. - _Michel Marcus_, Oct 21 2020

%H G. C. Greubel, <a href="/A025226/b025226.txt">Table of n, a(n) for n = 1..900</a>

%H Volkan Yildiz, <a href="https://arxiv.org/abs/2010.10303">Counting with 3-valued truth tables of bracketed formulae connected by implication</a>, arXiv:2010.10303 [math.GM], 2020.

%H Volkan Yildiz, <a href="https://arxiv.org/abs/2106.04728">Notes on algebraic structure of truth tables of bracketed formulae connected by implications</a>, arXiv:2106.04728 [math.CO], 2021.

%F a(n) = Sum_{j=1..n-1} a(j)*a(n-j), with a(1) = 3.

%F a(n) = 3^n * A000108(n-1).

%F G.f.: (1-sqrt(1-12*x))/2. - _Michael Somos_, Jun 08 2000

%F 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

%F 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

%F D-finite with recurrence: n*a(n) +6*(3-2*n)*a(n-1) = 0. - _R. J. Mathar_, Nov 12 2012

%F a(n) = 3^n/(4*n-2)*binomial(2*n,n). - _Vaclav Kotesovec_, Oct 11 2013

%e a(3) = 3^3*C(2) = 27*2 = 54.

%t Rest[CoefficientList[Series[(1-Sqrt[1-12x])/2,{x,0,20}],x]] (* _Harvey P. Dale_, Mar 09 2011 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-12*x+x*O(x^n)))/2,n)

%o (Magma) [3^n*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, May 20 2022

%o (SageMath) [3^n*catalan_number(n-1) for n in (1..30)] # _G. C. Greubel_, May 20 2022

%Y Cf. A000108, A005159.

%K nonn

%O 1,1

%A _Clark Kimberling_