A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003

Let P(A) be the power set of an n-element set A. Then a(n-2) is the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x. - Ross La Haye, Jan 10 2008

a(n) also gives the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011

a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• K. S. Brown, Dedekind's problem
• John Elias, Illustration of Initial Terms: Inverse of the Sierpinski Triangle
• Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
• Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92). - _N. J. A. Sloane_, May 12 2012
• Index entries for sequences related to Boolean functions
• Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Crossrefs

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.

Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

Programs

Magma

[(2^n)*(2^n-1)/2-3^n+2^n: n in [2..30]]; // Vincenzo Librandi, Oct 06 2017

Maple

a:= n-> Stirling2(n+4, 4)-Stirling2(n+3, 4): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 05 2007

Mathematica

CoefficientList[1/((1-2x)(1-3x)(1-4x)) + O[x]^30, x] (* Jean-François Alcover, Nov 28 2015 *)
LinearRecurrence[{9, -26, 24}, {1, 9, 55}, 40] (* Vincenzo Librandi, Oct 06 2017 *)

PARI

a(n)=(2^n)*(2^n-1)/2-3^n+2^n \\ Charles R Greathouse IV, Mar 22 2016

Formula

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).

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

From Hieronymus Fischer, Jun 25 2007: (Start)

a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.

a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)

a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 2008

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,2), (n >= 2). - Milan Janjic, Apr 26 2009

E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011

a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016

a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019