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A094033 Number of connected 2-element antichains on a labeled n-set.

%I A094033 #44 Jul 17 2024 16:56:28
%S A094033 0,0,0,3,18,75,270,903,2898,9075,27990,85503,259578,784875,2366910,
%T A094033 7125303,21425058,64373475,193317030,580344303,1741819338,5227030875,
%U A094033 15684238350,47059006503,141189602418,423593973075,1270832250870
%N A094033 Number of connected 2-element antichains on a labeled n-set.
%C A094033 Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) 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 10 2008
%H A094033 G. C. Greubel, <a href="/A094033/b094033.txt">Table of n, a(n) for n = 0..1000</a>
%H A094033 Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, <a href="https://arxiv.org/abs/2407.04409">The Fibonacci-Fubini and Lucas-Fubini numbers</a>, arXiv:2407.04409 [math.CO], 2024. See p. 10.
%H A094033 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 A094033 Adam Roman, Igor T. Podolak and Agnieszka Deszynska, <a href="https://www.ejournals.eu/sj/index.php/SI/article/view/2227">On the number of clusterings in a hierarchical classification model with overlapping clusters</a>, Schedae Informaticae, Volume 20, 2011.
%H A094033 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6, -11, 6).
%F A094033 a(n) = 3 * A000392(n).
%F A094033 E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
%F A094033 From _Colin Barker_, Mar 31 2012: (Start)
%F A094033 a(n) = (3^n-3*2^n+3)/2.
%F A094033 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
%F A094033 G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)
%p A094033 [seq(stirling2(n,3)*3,n=0..26)]; # _Zerinvary Lajos_, Dec 06 2006
%t A094033 Table[3 StirlingS2[n, 3], {n, 0, 26}] (* _Michael De Vlieger_, Nov 30 2015 *)
%o A094033 (PARI) x='x+O('x^50); concat([0,0,0],Vec(serlaplace((exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!))) \\ _G. C. Greubel_, Oct 06 2017
%Y A094033 Cf. A016269, A047707.
%Y A094033 Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.
%Y A094033 Cf. A094034, A094035, A094036, A094037.
%K A094033 nonn
%O A094033 0,4
%A A094033 Goran Kilibarda, _Vladeta Jovovic_, Apr 22 2004