A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

0, 1, 5, 22, 95, 406, 1715, 7162, 29615, 121486, 495275, 2009602, 8124935, 32761366, 131834435, 529712842, 2125993055, 8525430046, 34166159195, 136858084882, 548012945975, 2193794127526, 8780404589555, 35137304693722

Offset: 0

Author

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

Keywords

• easy
• nonn

Comments

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is 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, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 11 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
• V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Cf. A036239, A083324.

Cf. A000225, A032263, A028243.

Programs

Magma

[(4^n-3^n+2^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 06 2017

Mathematica

Table[(4^n-3^n+2^n-1)/2, {n,1,30}] (* Clark Kimberling, Mar 12 2012 *)
CoefficientList[Series[x (1 - 5 x + 7 x^2) / ((1 - x) (1 - 4 x) (1 - 3 x) (1 - 2 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 06 2017 *)

PARI

a(n) = (4^n-3^n+2^n-1)/2; \\ Michel Marcus, Nov 30 2015

Formula

a(n) = (A083324(n) - 1)/2.

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

a(n) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008

From Wolfdieter Lang, Oct 28 2011 (Start)

E.g.f.: Sum_{j=1..4} ((-1)^j*exp(j*x))/2 = exp(x)*(exp(4*x)-1)/(exp(x)+1)/2.

O.g.f.: Sum_{j=1..4} (((-1)^j)/(1-j*x))/2 = x*(1-5*x+7*x^2)/product(1-j*x,j=1..4). See A196847.

(End)

G.f.: x*(1-5*x+7*x^2)/((1-x)*(1-4*x)*(1-3*x)*(1-2*x)). - Vincenzo Librandi, Oct 06 2017