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A094033
Number of connected 2-element antichains on a labeled n-set.
16
0, 0, 0, 3, 18, 75, 270, 903, 2898, 9075, 27990, 85503, 259578, 784875, 2366910, 7125303, 21425058, 64373475, 193317030, 580344303, 1741819338, 5227030875, 15684238350, 47059006503, 141189602418, 423593973075, 1270832250870
OFFSET
0,4
COMMENTS
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
LINKS
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 10.
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.
Adam Roman, Igor T. Podolak and Agnieszka Deszynska, On the number of clusterings in a hierarchical classification model with overlapping clusters, Schedae Informaticae, Volume 20, 2011.
FORMULA
a(n) = 3 * A000392(n).
E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
From Colin Barker, Mar 31 2012: (Start)
a(n) = (3^n-3*2^n+3)/2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)
MAPLE
[seq(stirling2(n, 3)*3, n=0..26)]; # Zerinvary Lajos, Dec 06 2006
MATHEMATICA
Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)
PROG
(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
KEYWORD
nonn
AUTHOR
Goran Kilibarda, Vladeta Jovovic, Apr 22 2004
STATUS
approved