A047707 Number of monotone Boolean functions of n variables with 3 mincuts. Also Sperner systems with 3 blocks.

0, 0, 0, 2, 64, 1090, 14000, 153762, 1533504, 14356610, 128722000, 1119607522, 9528462944, 79817940930, 660876543600, 5424917141282, 44246078560384, 359144709794050, 2904688464582800, 23429048035827042, 188593339362097824

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The paper by G. Kilibarda, Enumeration of certain classes of antichains, Publications de l'Institut Mathematique, Nouvelle série, 97 (111) (2015), mentions many sequences, but since only very condensed formulas are given, it is hard to match them with entries in the OEIS. It would be nice to add this reference to all the sequences that it mentions. - N. J. A. Sloane, Jan 01 2016

Term a(1108) has 1000 decimal digits. - Michael De Vlieger, Jan 26 2016

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,3).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1107
• K. S. Brown, Dedekind's problem.
• Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
• G. Kilibarda, Enumeration of certain classes of antichains, Publications de l'Institut Mathematique, Nouvelle série, 97 (111) (2015), 69-87 DOI: 10.2298/PIM140406001K. See page 86, formula for alpha^hat(3,n).
• Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
• Index entries for sequences related to Boolean functions
• Index entries for linear recurrences with constant coefficients, signature (28,-315,1820,-5684,9072,-5760).

Crossrefs

Cf. A016269, A051112.

Programs

Mathematica

Table[Binomial[2^n, 3] - (6^n - 5^n - 4^n + 3^n), {n, 20}] (* or *)
CoefficientList[Series[-2 x^3 (36 x^2 - 4 x - 1)/((2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1) (8 x - 1)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 26 2016 *)

PARI

a(n)=binomial(2^n,3)-(6^n-5^n-4^n+3^n) \\ Charles R Greathouse IV, Apr 08 2016

Formula

a(n) = (2^n)*(2^n - 1)*(2^n - 2)/6 - (6^n - 5^n - 4^n + 3^n).

G.f.: -2*x^3*(36*x^2-4*x-1)/((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Jul 31 2012

a(n) = Binomial(2^n,3) - (6^n - 5^n - 4^n + 3^n). - Ross La Haye, Jan 26 2016