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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.

%I A133224 #34 Aug 04 2022 15:50:48
%S A133224 0,2,14,78,400,1960,9312,43232,197120,885888,3934720,17307136,
%T A133224 75509760,327182336,1409343488,6039920640,25770065920,109522223104,
%U A133224 463857647616,1958507577344,8246342451200
%N 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.
%C A133224 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]
%H A133224 Vincenzo Librandi, <a href="/A133224/b133224.txt">Table of n, a(n) for n = 0..300</a>
%H A133224 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A133224 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-52,96,-64).
%F A133224 a(n) = n*(2^(n-2) + 3*2^(2*n-3)).
%F A133224 G.f.: 2*x*(7*x^2-5*x+1) / ((2*x-1)^2*(4*x-1)^2). [_Colin Barker_, Dec 10 2012]
%F A133224 E.g.f.: exp(2*x)*(1 + 3*exp(2*x))*x. - _Stefano Spezia_, Aug 04 2022
%e A133224 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.
%t A133224 LinearRecurrence[{12,-52,96,-64},{0,2,14,78},30] (* _Harvey P. Dale_, Jan 24 2019 *)
%o A133224 (Magma) [n*(2^(n-2) + 3*2^(2*n-3)): n in [0..30]]; // _Vincenzo Librandi_, Jun 10 2011
%Y A133224 Cf. A027471, A002697, A082134, A212698.
%K A133224 nonn,easy
%O A133224 0,2
%A A133224 _Ross La Haye_, Dec 30 2007, Jan 03 2008