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

1,2

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.

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.

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 *)

Crossrefs

Cf. A192481, A000108.

Sequence in context: A307099 A048175 A288953 * A233537 A020132 A330042

Adjacent sequences: A192479 A192480 A192481 * A192483 A192484 A192485

Keyword

nonn

Author

Volkan Yildiz, Jul 01 2011

Status

approved