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%I #15 Dec 01 2017 09:04:51
%S 0,1,3,10,45,226,1113,5230,23565,102826,438273,1836550,7601685,
%T 31183426,127084233,515429470,2083077405,8396552026,33779262993,
%U 135696871990,544528258725,2183337968626,8749031918553,35043178292110,140313885993645,561679104393226
%N 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 disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x.
%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. [_Ross La Haye_, Feb 22 2009]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F a(n) = (1/2)(4^n - 3^(n+1) + 5*2^n - 3) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,2).
%F G.f.: x*(1-7*x+15*x^2)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [_Colin Barker_, Jul 29 2012]
%e a(3) = 10 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we have for case 0 {{},{1}}, {{},{2}}, {{},{3}}, {{},{1,2}}, {{},{1,3}}, {{},{2,3}}, {{},{1,2,3}} and we have for case 1 {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}.
%t LinearRecurrence[{10,-35,50,-24},{0,1,3,10},30] (* _Harvey P. Dale_, Dec 01 2017 *)
%Y Cf. A000225, A032263.
%K nonn,easy
%O 0,3
%A _Ross La Haye_, Jan 10 2008
%E More terms from _Harvey P. Dale_, Dec 01 2017
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