A134064 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 intersecting but 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, or 2) x = y.
1, 2, 6, 23, 96, 407, 1716, 7163, 29616, 121487, 495276, 2009603, 8124936, 32761367, 131834436, 529712843, 2125993056, 8525430047, 34166159196, 136858084883, 548012945976, 2193794127527, 8780404589556, 35137304693723, 140596281975696, 562526325893807, 2250528914325516
Offset: 0
Examples
a(2) = 6 because for P(A) = {{},{1},{2},{1,2}} we have for case 1 {{1},{1,2}}, {{2},{1,2}} and we have for case 2 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 0.
Links
- 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},{1,2,6,23},30] (* Harvey P. Dale, Jul 04 2023 *) -
PARI
Vec((1-8*x+21*x^2-17*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Oct 30 2015
Formula
a(n) = (1/2)(4^n - 3^n + 2^n + 1) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1.
a(n) = C(2^n + 1,2) - (1/2)(3^n - 1) = StirlingS2(2^n + 1,2^n) - StirlingS2(n+1,3) - StirlingS2(n+1,2). - Ross La Haye, Jan 21 2008
G.f.: (1-8*x+21*x^2-17*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). - Colin Barker, Jul 30 2012