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A053154 Number of 2-element intersecting families (with not necessary distinct sets) of an n-element set. 7
0, 1, 5, 22, 95, 406, 1715, 7162, 29615, 121486, 495275, 2009602, 8124935, 32761366, 131834435, 529712842, 2125993055, 8525430046, 34166159195, 136858084882, 548012945975, 2193794127526, 8780404589555, 35137304693722 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

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 *)

PROG

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

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

CROSSREFS

Cf. A036239, A083324.

Cf. A000225, A032263, A028243.

Sequence in context: A026877 A128746 A049675 * A141222 A127360 A116415

Adjacent sequences:  A053151 A053152 A053153 * A053155 A053156 A053157

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

STATUS

approved

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)