A218185 - OEIS

%I #35 Oct 08 2016 10:10:03

%S 0,0,1,4,19,100,566,3384,21107,136084,900674,6087496,41850366,

%T 291766952,2057964492,14659421040,105305580483,761981900724,

%U 5548736343434,40632122219688,299017702596554,2210275626304248,16403005547059508,122169144755555088,912887876722311406

%N Number of rows with the value true in the truth tables of all bracketed formulas with n distinct propositions p_1, ..., p_n connected by the binary connective of m-implication (case 3).

%H Volkan Yildiz, <a href="http://arxiv.org/abs/1205.5595">General combinatorical structure of truth tables of bracketed formulas connected by implication</a>, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

%F Yildiz gives a g.f. (see Proposition 6.13).

%F G.f.: (-1 - sqrt(1-8*x) + sqrt(1-8*x)*sqrt(1-4*x) + sqrt(1-4*x) + 4*x)/4.

%F a(n+1) = Sum_{i=1..n} (C(i-1)*2^i*C(n-i)) - C(n), n>0. a(0)=0, C(n)=A000108(n) - Catalan numbers.   [_Vladimir Kruchinin_, Mar 20 2013]

%F Conjecture: n*(n-1)*(n-2)*a(n) -12*(2*n-5)*(n-1)*(n-2)*a(n-1) +4 *(n-2)*(52*n^2-312*n+465) *a(n-2) -48*(2*n-7)*(8*n^2-56*n+95)*a(n-3) +256*(n-5) *(2*n-7) *(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Oct 08 2016

%F Let h(n) = Catalan(n)*(2^(n+1)-1-hypergeom([3/2,1-n],[+1/2-n], 2))/2 then a(n+1) = h(n) for n>=1. - _Peter Luschny_, Oct 08 2016

%p h := n -> (binomial(2*n,n)/(n+1))*(2^(n+1)-1-hypergeom([3/2,1-n],[1/2-n],2))/2:

%p a := n -> `if`(n<2, 0, simplify(h(n-1))):

%p seq(a(n), n=0..24); # _Peter Luschny_, Oct 08 2016

%t Table[Boole[n > 1] (Sum[CatalanNumber[i - 1] 2^i*CatalanNumber[# - i], {i, 1, #}] - CatalanNumber@ # &[n - 1]), {n, 0, 24}] (* or *)

%t CoefficientList[Series[(-1 - Last@ # + Times @@ # + First@ # + 4 x)/4 &@ {Sqrt[1 - 4 x], Sqrt[1 - 8 x]}, {x, 0, 24}], x] (* _Michael De Vlieger_, Oct 08 2016 *)

%o (PARI) all_a(m) = {x= y+O(y^(m+1)); P = (-1 - sqrt(1-8*x) + sqrt(1-8*x)*sqrt(1-4*x) + sqrt(1-4*x) + 4*x)/4; for (n=0, m, print1(polcoeff(P, n, y), ", "));} \\ _Michel Marcus_, Feb 08 2013

%o (Maxima)

%o c(n):=binomial(2*n,n)/(n+1);

%o a(n):=sum(c(i-1)*2^i*c(n-i-1),i,1,n-1)-c(n-1);

%o makelist(a(n),n,2,20); /* _Vladimir Kruchinin_, Mar 20 2013 */

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Oct 23 2012

%E More terms from _Michel Marcus_, Feb 08 2013