A133224 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Remove (y,x) from B when (x,y) is in B and x <> y and let R35 denote the reduced set B. Then a(n) = the sum of the sizes of the union of x and y for every (x,y) in R35.

0, 2, 14, 78, 400, 1960, 9312, 43232, 197120, 885888, 3934720, 17307136, 75509760, 327182336, 1409343488, 6039920640, 25770065920, 109522223104, 463857647616, 1958507577344, 8246342451200

Offset

0,2

Comments

A082134 is the analogous sequence if "union" is replaced by "intersection" and A002697 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y = Y union X are considered as the same Cartesian product [Relation (37): U_Q(n) in document of Ross La Haye in reference], if we want to consider that X Union Y and Y Union X are two distinct Cartesian products, see A212698. [Bernard Schott, Jan 11 2013]

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

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 (12,-52,96,-64).

Formula

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

G.f.: 2*x*(7*x^2-5*x+1) / ((2*x-1)^2*(4*x-1)^2). [Colin Barker, Dec 10 2012]

E.g.f.: exp(2*x)*(1 + 3*exp(2*x))*x. - Stefano Spezia, Aug 04 2022

Example

a(2) = 14 because for P(A) = {{},{1},{2},{1,2}} |{} union {1}| = 1, |{} union {2}| = 1, |{} union {1,2}| = 2, |{1} union {2}| = 2, |{1} union {1,2}| = 2 and |{2} union {1,2}| = 2, |{} union {}| = 0, |{1} union {1}| = 1, |{2} union {2}| = 1, |{1,2} union {1,2}| = 2, which sums to 14.

Mathematica

LinearRecurrence[{12, -52, 96, -64}, {0, 2, 14, 78}, 30] (* Harvey P. Dale, Jan 24 2019 *)

Prog

(Magma) [n*(2^(n-2) + 3*2^(2*n-3)): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

Crossrefs

Cf. A027471, A002697, A082134, A212698.

Sequence in context: A034573 A278417 A339240 * A183577 A121200 A112408

Adjacent sequences: A133221 A133222 A133223 * A133225 A133226 A133227

Keyword

nonn,easy

Author

Ross La Haye, Dec 30 2007, Jan 03 2008

Status

approved