%I #38 Sep 08 2022 08:45:10
%S 1,2,6,20,66,212,666,2060,6306,19172,58026,175100,527346,1586132,
%T 4766586,14316140,42981186,129009092,387158346,1161737180,3485735826,
%U 10458256052,31376865306,94134790220,282412759266,847255055012
%N a(n) = 3^n - 2^n + 1.
%C Binomial transform of A000225 (if this starts 1,1,3,7....).
%C 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 intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 1) x = y. - _Ross La Haye_, Jan 10 2008
%C Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - _Ross La Haye_, Mar 19 2009
%H M. H. Albert, M. D. Atkinson, and V. Vatter, <a href="http://arxiv.org/abs/1209.0425">Inflations of geometric grid classes: three case studies</a>, arXiv:1209.0425 [math.CO], 2012.
%H 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 Jay Pantone, <a href="http://arxiv.org/abs/1309.0832">The Enumeration of Permutations Avoiding 3124 and 4312</a>, arXiv:1309.0832 [math.CO], 2013.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).
%F E.g.f.: exp(3*x) - exp(2*x) + exp(x).
%F Row sums of triangle A134319. - _Gary W. Adamson_, Oct 19 2007
%F a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - _Ross La Haye_, Jan 10 2008
%F a(n) = Sum_{k=0..n}(binomial(n,k)*A255047(k)). - _Yuchun Ji_, Feb 23 2019
%e From _Gus Wiseman_, Dec 10 2019: (Start)
%e Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
%e 0 0 0 0
%e {1} {1} {1}
%e {2} {2}
%e {12} {3}
%e {1}{2} {12}
%e {1}{2}{12} {13}
%e {23}
%e {123}
%e {1}{2}
%e {1}{3}
%e {2}{3}
%e {1}{2}{3}
%e {1}{2}{12}
%e {1}{3}{13}
%e {2}{3}{23}
%e {12}{13}{23}
%e {1}{2}{3}{123}
%e {12}{13}{23}{123}
%e {1}{2}{3}{12}{13}{23}
%e {1}{2}{3}{12}{13}{23}{123}
%e BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229.
%e (End)
%t LinearRecurrence[{6,-11,6}, {1,2,6}, 30] (* _G. C. Greubel_, Feb 13 2019 *)
%o (PARI) a(n)=3^n-2^n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [3^n-2^n+1: n in [0..30]]; // _G. C. Greubel_, Feb 13 2019
%o (Sage) [3^n-2^n+1 for n in range(30)] # _G. C. Greubel_, Feb 13 2019
%o (GAP) List([0..30], n -> 3^n-2^n+1); # _G. C. Greubel_, Feb 13 2019
%Y Cf. A134319, A028243, A000079.
%Y Cf. A000612, A003238, A330098, A330234.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Apr 27 2003