A192482 - OEIS

%I #30 Feb 28 2017 22:54:51

%S 1,3,10,51,286,1710,10740,69763,464822,3159450,21821516,152708078,

%T 1080452972,7716009724,55545950568,402649640163,2936600795174,

%U 21532660592418,158645924209500,1173875395710458,8719519396134596,64995349923442628,486020221692290392

%N 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.

%C 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).

%H Alois P. Heinz, <a href="/A192482/b192482.txt">Table of n, a(n) for n = 1..500</a>

%H Volkan Yildiz, <a href="http://arxiv.org/abs/1203.4645">Counting false entries in truth tables of bracketed formulas connected by m-implication</a>, arXiv:1203.4645 [math.CO], 2012.

%F 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.

%p C:= n-> binomial(2*n, n)/(n+1):

%p y:= proc(n) option remember;

%p       `if`(n<2, n, add((2^i    *C(i-1)  -y(i))*

%p                        (2^(n-i)*C(n-i-1)-y(n-i)), i=1..n-1))

%p     end:

%p a:= n-> 2^n*C(n-1) -y(n):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Feb 06 2012

%t 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_ *)

%Y Cf. A192481, A000108.

%K nonn

%O 1,2

%A _Volkan Yildiz_, Jul 01 2011