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%I #38 Jan 29 2023 19:31:36
%S 0,1,5,22,95,406,1715,7162,29615,121486,495275,2009602,8124935,
%T 32761366,131834435,529712842,2125993055,8525430046,34166159195,
%U 136858084882,548012945975,2193794127526,8780404589555,35137304693722
%N Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.
%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 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, or 2) 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 11 2008
%H G. C. Greubel, <a href="/A053154/b053154.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Jovovic, G. Kilibarda, <a href="http://dx.doi.org/10.4213/dm398">On the number of Boolean functions in the Post classes F^{mu}_8</a>, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
%H V. Jovovic, G. Kilibarda, <a href="http://dx.doi.org/10.1515/dma.1999.9.6.593">On the number of Boolean functions in the Post classes F^{mu}_8</a>, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F a(n) = (A083324(n) - 1)/2.
%F a(n) = (4^n - 3^n + 2^n - 1)/2.
%F a(n) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2). - _Ross La Haye_, Jan 11 2008
%F From _Wolfdieter Lang_, Oct 28 2011 (Start)
%F E.g.f.: Sum_{j=1..4} ((-1)^j*exp(j*x))/2 = exp(x)*(exp(4*x)-1)/(exp(x)+1)/2.
%F O.g.f.: Sum_{j=1..4} (((-1)^j)/(1-j*x))/2 = x*(1-5*x+7*x^2)/product(1-j*x,j=1..4). See A196847.
%F (End)
%F G.f.: x*(1-5*x+7*x^2)/((1-x)*(1-4*x)*(1-3*x)*(1-2*x)). - _Vincenzo Librandi_, Oct 06 2017
%t Table[(4^n-3^n+2^n-1)/2, {n,1,30}] (* _Clark Kimberling_, Mar 12 2012 *)
%t CoefficientList[Series[x (1 - 5 x + 7 x^2) / ((1 - x) (1 - 4 x) (1 - 3 x) (1 - 2 x)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Oct 06 2017 *)
%o (PARI) a(n) = (4^n-3^n+2^n-1)/2; \\ _Michel Marcus_, Nov 30 2015
%o (Magma) [(4^n-3^n+2^n-1)/2: n in [0..30]]; // _Vincenzo Librandi_, Oct 06 2017
%Y Cf. A036239, A083324.
%Y Cf. A000225, A032263, A028243.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Goran Kilibarda, Feb 28 2000
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