A066524 a(n) = n*(2^n - 1).

0, 1, 6, 21, 60, 155, 378, 889, 2040, 4599, 10230, 22517, 49140, 106483, 229362, 491505, 1048560, 2228207, 4718574, 9961453, 20971500, 44040171, 92274666, 192937961, 402653160, 838860775, 1744830438, 3623878629, 7516192740, 15569256419, 32212254690

Offset: 0

Author

Henry Bottomley, Jan 08 2002

Keywords

• nonn
• easy

Comments

a(n)/2^n is the expected value of the cardinality of the generalized union of n randomly selected (with replacement) subsets of [n] where the probability of selection is equal for all subsets. - Geoffrey Critzer, May 18 2009

Form a triangle in which interior members T(i,j) = T(i-1,j-1) + T(i-1,j). The exterior members are given by 1,2,3,...,2*n-1: T(1,1) = n, T(2,1) = n-1, T(3,1) = n-2, ..., T(n,1) = 1 and T(2,2) = n + 1, T(3,3) = n + 2, ..., T(n,n) = 2*n - 1. The sum of all members will reproduce this sequence. For example, with n = 4 the exterior members are 1 to 7: row(1) = 4; row(2) = 3,5; row(3) = 2,8,6; row(4) = 1,10,14,7. The sum of all these members is 60, the fourth term in the sequence. - J. M. Bergot, Oct 16 2012

Examples

			a(4) = 4*(2^4 - 1) = 4*15 = 60.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..250
• A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A000225, A132751, A335764.

Programs

Magma

[n*(2^n-1): n in [0..30]]; // Vincenzo Librandi, Jan 24 2016

Mathematica

Table[n*2^n-n,{n,0,3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
CoefficientList[Series[x (1 - 2 x^2)/((1 - x) (1 - 2 x))^2, {x, 0, 30}], x] (* Michael De Vlieger, Jan 24 2016 *)

Sage

[gaussian_binomial(n,1,2)*n for n in range(0,29)] # Zerinvary Lajos, May 29 2009

Formula

a(n) = 2*a(n-1) + 2^n = A000225(n) * A001477(n) = A036289(n) - A001477(n).

G.f.: x*(1 - 2*x^2)/((1 - x)*(1 - 2*x))^2.

a(n) = n * Sum_{j = 1..n} binomial(n,j), n >= 0. - Zerinvary Lajos, May 10 2007

Row sums of triangles A132751. - Gary W. Adamson, Aug 28 2007

E.g.f.: x*(2*exp(2*x) - exp(x)). From an earlier rewritten comment. - Wolfdieter Lang, Feb 16 2016

Sum_{n>=1} 1/a(n) = A335764. - Amiram Eldar, Jun 23 2020