This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A025226 #37 May 21 2022 08:29:30
%S A025226 3,9,54,405,3402,30618,288684,2814669,28146690,287096238,2975361012,
%T A025226 31241290626,331638315876,3553267670100,38375290837080,
%U A025226 417331287853245,4566095267100210
%N 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).
%C A025226 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 A025226 G. C. Greubel, <a href="/A025226/b025226.txt">Table of n, a(n) for n = 1..900</a>
%H A025226 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 A025226 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 A025226 a(n) = Sum_{j=1..n-1} a(j)*a(n-j), with a(1) = 3.
%F A025226 a(n) = 3^n * A000108(n-1).
%F A025226 G.f.: (1-sqrt(1-12*x))/2. - _Michael Somos_, Jun 08 2000
%F A025226 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 A025226 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 A025226 D-finite with recurrence: n*a(n) +6*(3-2*n)*a(n-1) = 0. - _R. J. Mathar_, Nov 12 2012
%F A025226 a(n) = 3^n/(4*n-2)*binomial(2*n,n). - _Vaclav Kotesovec_, Oct 11 2013
%e A025226 a(3) = 3^3*C(2) = 27*2 = 54.
%t A025226 Rest[CoefficientList[Series[(1-Sqrt[1-12x])/2,{x,0,20}],x]] (* _Harvey P. Dale_, Mar 09 2011 *)
%o A025226 (PARI) a(n)=polcoeff((1-sqrt(1-12*x+x*O(x^n)))/2,n)
%o A025226 (Magma) [3^n*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, May 20 2022
%o A025226 (SageMath) [3^n*catalan_number(n-1) for n in (1..30)] # _G. C. Greubel_, May 20 2022
%Y A025226 Cf. A000108, A005159.
%K A025226 nonn
%O A025226 1,1
%A A025226 _Clark Kimberling_