A192482 a(n) = 2^n*C(n-1)-y(n), where y(n) = Sum_{i=1..n-1} (2^i*C(i-1)-y(i))*(2^(n-i)*C(n-i-1)-y(n-i)), y(0)=0, y(1)=1 and where C(i) is the i-th Catalan number.

1, 3, 10, 51, 286, 1710, 10740, 69763, 464822, 3159450, 21821516, 152708078, 1080452972, 7716009724, 55545950568, 402649640163, 2936600795174, 21532660592418, 158645924209500, 1173875395710458, 8719519396134596, 64995349923442628, 486020221692290392

Offset: 1

Author

Volkan Yildiz, Jul 01 2011

Keywords

• nonn

Comments

The sequence a(n) for n>=1 is the number of rows with the value true in the truth tables of all bracketed formulas with n distinct variables connected by the binary connective of m-implication, case(i).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• Volkan Yildiz, Counting false entries in truth tables of bracketed formulas connected by m-implication, arXiv:1203.4645 [math.CO], 2012.

Crossrefs

Cf. A192481, A000108.

Programs

Maple

C:= n-> binomial(2*n, n)/(n+1):
y:= proc(n) option remember;
      `if`(n<2, n, add((2^i    *C(i-1)  -y(i))*
                       (2^(n-i)*C(n-i-1)-y(n-i)), i=1..n-1))
    end:
a:= n-> 2^n*C(n-1) -y(n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 06 2012

Mathematica

c = CatalanNumber; y[n_] := y[n] = If[n<2, n, Sum[(2^i*c[i-1]-y[i])*(2^(n-i)*c[n-i-1] - y[n-i]), {i, 1, n-1}]]; a[n_] := 2^n*c[n-1]-y[n]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

Formula

a(n) = 2^n*C(n-1)-y(n), where y(n) = Sum_{i=1..n-1} (2^i*C(i-1)-y(i))*(2^(n-i)*C(n-i-1)-y(n-i)), y(0)=0, y(1)=1 and C(i) is the i-th Catalan number.