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

1,2

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

Table of n, a(n) for n=1..21.

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

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

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 *)

Crossrefs

Cf. A186997.

Sequence in context: A228251 A348200 A218092 * A161799 A378889 A378828

Adjacent sequences: A192476 A192477 A192478 * A192480 A192481 A192482

Keyword

nonn

Author

Volkan Yildiz, Jul 01 2011

Status

approved