A192479 a(n) = 2^n*C(n-1) - A186997(n-1), where C(n) are the Catalan numbers (A000108).

1, 3, 12, 61, 344, 2074, 13080, 85229, 569264, 3876766, 26817304, 187908802, 1330934032, 9513485076, 68539442800, 497178707325, 3628198048352, 26617955242806, 196205766112536, 1452410901340598, 10792613273706320

Offset: 1

Author

Volkan Yildiz, Jul 01 2011

Keywords

• nonn

Comments

a(n) is the number of rows with the value true in the truth tables of all bracketed formulas with n distinct propositions connected by the binary connective of implication.

Links

• P. J. Cameron and V. Yildiz, Counting false entries in truth tables of bracketed formulas connected by implication, arXiv:1106.4443 [math.CO], 2011.

Crossrefs

Cf. A186997.

Programs

Maple

C := proc(n) binomial(2*n,n)/(n+1) ;end proc:
Yildf := proc(n) option remember; if n<=1 then 1; else add( (2^i*C(i-1)-procname(i))*procname(n-i),i=1..n-1) ; end if; end proc:
A192479 := proc(n) 2^n*C(n-1)-Yildf(n) ; end proc:
seq(A192479(n),n=1..30) ; # R. J. Mathar, Jul 13 2011

Mathematica

a[1] = 1; a[n_] := 2^n*CatalanNumber[n-1] - Sum[Binomial[k, n-k-1]*Binomial[n+2*k-1, n+k-1]/(n+k), {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2015 *)

Formula

a(n) = 2^n*C(n) - f(n), with f(n) = Sum_{i=1..n-1} (2^i*C(i)-f(i))*f(n-i), starting f(0)=f(1)=1, where C(i) = A000108(i-1).

G.f.: 1 - 1/A186997(x). - Vladimir Kruchinin, Feb 17 2013

a(n+1) = Sum_{k=1..n+1} (binomial(k,n-k+1)*binomial(n+2*k-1,k))/(n+k), a(1)=1. - Vladimir Kruchinin, May 15 2014