A133789 - cp's OEIS frontend

A133789 Let P(A) denote 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 disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.

%I A133789 #26 May 15 2017 18:24:04
%S A133789 0,1,4,16,70,316,1414,6196,26590,112156,466774,1923076,7863310,
%T A133789 31972396,129459334,522571156,2104535230,8460991036,33972711094,
%U A133789 136277478436,546270602350,2188566048076,8764718254054,35090241492916,140455083984670,562102715143516
%N A133789 Let P(A) denote 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 disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.
%C A133789 Also, number of even binomial coefficient in rows 0 to 2^n of Pascal's triangle. [_Aaron Meyerowitz_, Oct 29 2013]
%H A133789 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. [_Ross La Haye_, Feb 22 2009]
%F A133789 a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
%F A133789 O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - _R. J. Mathar_, Jan 11 2008
%F A133789 a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - _Vladeta Jovovic_, Jan 04 2008, corrected by _Eric Rowland_, May 15 2017
%F A133789 a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - _Ross La Haye_, Jan 11 2008
%F A133789 a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - _Ross La Haye_, Jan 11 2008
%F A133789 a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [_Aaron Meyerowitz_, Oct 29 2013]
%e A133789 a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
%e A133789 {1} and {2},
%e A133789 {1} and {3},
%e A133789 {2} and {3},
%e A133789 {1} and {2,3},
%e A133789 {2} and {1,3},
%e A133789 {3} and {1,2}
%e A133789 are disjoint, while
%e A133789 {} and {1},
%e A133789 {} and {2},
%e A133789 {} and {3},
%e A133789 {} and {1,2},
%e A133789 {} and {1,3},
%e A133789 {} and {2,3},
%e A133789 {} and {1,2,3}
%e A133789 are disjoint and one is a subset of the other and
%e A133789 {1,2} and {1,3},
%e A133789 {1,2} and {2,3},
%e A133789 {1,3} and {2,3}
%e A133789 are intersecting, but neither is a subset of the other.
%e A133789 Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then  6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]
%Y A133789 Cf. A082134, A002697, A020522, A006516, A007582, A000302.
%Y A133789 Cf. A000225, A000392, A032263.
%K A133789 nonn
%O A133789 0,3
%A A133789 _Ross La Haye_, Jan 03 2008, Jan 08 2008
%E A133789 Edited by _N. J. A. Sloane_, Jan 20 2008 to incorporate suggestions from several contributors.

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