

A066524


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


14



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
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OFFSET

0,3


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(i1,j1) + T(i1,j). The exterior members are given by 1,2,3,...2*n1: T(1,1) = n, T(2,1) = n1, T(3,1) = n2, ..., 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


LINKS

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


FORMULA

a(n) = 2*a(n1) + 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} C(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


EXAMPLE

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


MATHEMATICA

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


PROG

(Sage) [gaussian_binomial(n, 1, 2)*n for n in range(0, 29)] # Zerinvary Lajos, May 29 2009
(PARI) { for (n=0, 250, write("b066524.txt", n, " ", n*(2^n  1)) ) } \\ Harry J. Smith, Feb 21 2010
(MAGMA) [n*(2^n1): n in [0..30]]; // Vincenzo Librandi, Jan 24 2016


CROSSREFS

Cf. A000225, A132751, A335764.
Sequence in context: A258142 A321257 A305120 * A113070 A009147 A012593
Adjacent sequences: A066521 A066522 A066523 * A066525 A066526 A066527


KEYWORD

nonn,easy


AUTHOR

Henry Bottomley, Jan 08 2002


STATUS

approved



