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A134057 a(n) = binomial(2^n-1,2). 3
0, 0, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 2094081, 8382465, 33542145, 134193153, 536821761, 2147385345, 8589737985, 34359345153, 137438167041, 549754241025, 2199020109825, 8796086730753, 35184359505921 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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 x is not a subset of y and y is not 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: Number of connections between the nodes of the perfect depth n binary tree and the nodes of a perfect depth (n-1) binary tree. - Alex Ratushnyak, Jun 02 2013

LINKS

Robert Israel, Table of n, a(n) for n = 0..1600

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 (7, -14, 8).

FORMULA

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

a(n) = 3 *A006095(n).

a(n) = (2^n-1)*(2^(n-1)-1). - Alex Ratushnyak, Jun 02 2013

a(n) = StirlingS2(2^n - 1,2^n - 2).

G.f.: 3*x^2/(1-x)/(1-2*x)/(1-4*x). - Colin Barker, Feb 22 2012

a(n) = A000225(n)*A000225(n-1). - Michel Marcus, Nov 30 2015

a(n) = A000217(2^n-2). - Michel Marcus, Nov 30 2015

EXAMPLE

a(2) = 3 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{1},{2}} and we have for case 2 {{1},{1,2}}, {{2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 1.

MAPLE

seq((2^n-1)*(2^(n-1)-1), n=0..100); # Robert Israel, Nov 30 2015

MATHEMATICA

Table[Binomial[2^n - 1, 2], {n, 0, 30}] (* Vincenzo Librandi, Nov 30 2015 *)

PROG

(Python)

for n in range(33):  print int((2**n-1)*(2**(n-1)-1)),  # Alex Ratushnyak, Jun 02 2013

(PARI) a(n) = binomial(2^n-1, 2); \\ Michel Marcus, Nov 30 2015

(MAGMA) [Binomial(2^n-1, 2): n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

CROSSREFS

Cf. A000217, A000225, A000392, A032263, A028243.

Sequence in context: A289399 A306093 A076207 * A128281 A034268 A140451

Adjacent sequences:  A134054 A134055 A134056 * A134058 A134059 A134060

KEYWORD

nonn,easy

AUTHOR

Ross La Haye, Jan 11 2008, Jun 01 2008

STATUS

approved

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Last modified August 20 05:25 EDT 2019. Contains 326139 sequences. (Running on oeis4.)