A134168 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, or 3) x = y.
1, 3, 9, 30, 111, 438, 1779, 7290, 29871, 121998, 496299, 2011650, 8129031, 32769558, 131850819, 529745610, 2126058591, 8525561118, 34166421339, 136858609170, 548013994551, 2193796224678, 8780408783859, 35137313082330, 140596298752911, 562526359448238, 2250528981434379, 9003386657325090
Offset: 0
Examples
a(2) = 9 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 2 {{1},{1,2}}, {{2},{1,2}} and we have for case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- 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.
- Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
Programs
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Mathematica
LinearRecurrence[{10, -35, 50, -24}, {1, 3, 9, 30}, 50] (* or *) Table[(1/2)*(4^n - 3^n + 3*2^n - 1), {n,0,50}] (* G. C. Greubel, May 30 2016 *)
Formula
a(n) = (1/2)*(4^n - 3^n + 3*2^n - 1).
a(n) = 3*StirlingS2(n+1,4) +2*StirlingS2(n+1,3) +2*StirlingS2(n+1,2) +1.
G.f.: -(5*x^3 - 14*x^2 + 7*x - 1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Jul 30 2012