A134057 a(n) = binomial(2^n-1,2).

0, 0, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 2094081, 8382465, 33542145, 134193153, 536821761, 2147385345, 8589737985, 34359345153, 137438167041, 549754241025, 2199020109825, 8796086730753, 35184359505921

Offset: 0

Author

Ross La Haye, Jan 11 2008, Jun 01 2008

Keywords

• nonn
• easy

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

a(n) is the number of positive entries in the positive rows and columns of a Walsh matrix of order 2^n. It is also the size of the smallest nontrivial conjugacy class in the general linear group GL(n,2). See the link "3-bit Walsh permutation...". - Tilman Piesk, Sep 15 2022

Examples

			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.
		

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
• Tilman Piesk, 3-bit Walsh permutation; conjugacy class 2+2 (Wikiversity)
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A000217, A000225, A000392, A032263, A028243, A171477.

Programs

Magma

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

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 *)

PARI

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

Python

print([(2**n-1)*(2**(n-1)-1) for n in range(23)])
# Alex Ratushnyak, Jun 02 2013

Formula

a(n) = (1/2)*(4^n - 3*2^n + 2) = 3*(Stirling2(n+1,4) + Stirling2(n+1,3)).

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

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

a(n) = Stirling2(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

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Wesley Ivan Hurt, May 17 2021

E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 2)/2. - Stefano Spezia, Apr 06 2022