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).

3, 9, 54, 405, 3402, 30618, 288684, 2814669, 28146690, 287096238, 2975361012, 31241290626, 331638315876, 3553267670100, 38375290837080, 417331287853245, 4566095267100210

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