A218185 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).

0, 0, 1, 4, 19, 100, 566, 3384, 21107, 136084, 900674, 6087496, 41850366, 291766952, 2057964492, 14659421040, 105305580483, 761981900724, 5548736343434, 40632122219688, 299017702596554, 2210275626304248, 16403005547059508, 122169144755555088, 912887876722311406

Offset

0,4

Links

Table of n, a(n) for n=0..24.

Volkan Yildiz, General combinatorical structure of truth tables of bracketed formulas connected by implication, arXiv preprint arXiv:1205.5595 [math.CO], 2012.

Formula

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

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

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]

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

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

Maple

h := n -> (binomial(2*n, n)/(n+1))*(2^(n+1)-1-hypergeom([3/2, 1-n], [1/2-n], 2))/2:
a := n -> `if`(n<2, 0, simplify(h(n-1))):
seq(a(n), n=0..24); # Peter Luschny, Oct 08 2016

Mathematica

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

Prog

(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
(Maxima)
c(n):=binomial(2*n, n)/(n+1);
a(n):=sum(c(i-1)*2^i*c(n-i-1), i, 1, n-1)-c(n-1);
makelist(a(n), n, 2, 20); /* Vladimir Kruchinin, Mar 20 2013 */

Crossrefs

Sequence in context: A177249 A083882 A007564 * A369672 A086624 A307121

Adjacent sequences: A218182 A218183 A218184 * A218186 A218187 A218188

Keyword

nonn

Author

N. J. A. Sloane, Oct 23 2012

Extensions

More terms from Michel Marcus, Feb 08 2013

Status

approved