A134045 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 = y.

1, 3, 7, 18, 61, 258, 1177, 5358, 23821, 103338, 439297, 1838598, 7605781, 31191618, 127100617, 515462238, 2083142941, 8396683098, 33779525137, 135697396278, 544529307301, 2183340065778, 8749036112857, 35043186680718, 140313902770861, 561679137947658, 2247987249823777, 8995761328275558

Offset

0,2

Links

Table of n, a(n) for n=0..27.

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. [Ross La Haye, Feb 22 2009]

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Formula

a(n) = (1/2)(4^n - 3^(n+1) + 7*2^n - 3) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,2) + 1.

G.f.: (1-7*x+12*x^2+3*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [Colin Barker, Jul 29 2012]

Example

a(2) = 7 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},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.

Mathematica

Table[3 StirlingS2[n + 1, 4] + 2 StirlingS2[n + 1, 2] + 1, {n, 0, 27}] (* Michael De Vlieger, Nov 30 2015 *)

Prog

(PARI) a(n) = (4^n - 3^(n+1) + 7*2^n - 3)/2; \\ Michel Marcus, Nov 30 2015

Crossrefs

Cf. A000225, A032263, A000079.

Sequence in context: A366578 A259885 A110578 * A079898 A173449 A270519

Adjacent sequences: A134042 A134043 A134044 * A134046 A134047 A134048

Keyword

nonn,easy

Author

Ross La Haye, Jan 11 2008

Extensions

More terms from Michael De Vlieger, Nov 30 2015

Status

approved