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A077585 a(n) = 2^(2^n-1) - 1. 10
0, 1, 7, 127, 32767, 2147483647, 9223372036854775807, 170141183460469231731687303715884105727, 57896044618658097711785492504343953926634992332820282019728792003956564819967 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(2), a(3), a(5) and a(7) are prime; a(11) is not.

Let S be a set of n elements. First we perform a set partition on S. Let SU be the set of all nonempty subsets of S. As is well known, 2^n - 1 is the number of nonempty subsets of a set with n elements (see A000225). That is, 2^n - 1 is the number of elements |SU| of SU. In the second step, we select k elements from SU. We want to know how many different selections are possible. Let W be the resulting set of selections formed from SU. Then the number of elements |W| of W is |W| = sum((binomial(2^n-1,k)), k=1..2^n-1) = 2^(2^n-1)-1 = A077585. Example: |W(n)| = a(n=2) = 7, because W={[[1, 2]], [[1]], [[1, 2], [1]], [[1, 2], [2], [1]], [[1, 2], [2]], [[2], [1]], [[2]]}. - Thomas Wieder, Nov 08 2007

LINKS

Table of n, a(n) for n=0..8.

Eric Weisstein's World of Mathematics, Double Mersenne Number

FORMULA

a(n) = A000225(A000225(n)).

a(n) = A058891(n+1)-1. - corrected by Maurizio De Leo, Feb 25 2015

a(n) = (A001146(n)-2)/2.

a(n) = A056220(1+a(n-1)).

a(n) = sum((binomial(2^n-1,k)), k=1..2^n-1). - Thomas Wieder, Nov 08 2007

a(n) = 2 * a(n-1)^2 + 4 * a(n-1) + 1. - Roderick MacPhee, Oct 05 2012

EXAMPLE

a(5) = 2^(2^5-1)-1 = 2^31-1 = 2147483647.

MAPLE

ZahlDerAuswahlenAusMengeDerZerlegungenEinerMenge:=proc() local n, nend, arg, k, w; nend:=10; for n from 1 to nend do arg:=2^n-1; w[n]:=sum((binomial(arg, k)), k=1..arg); od; print(w[1], w[2], w[3], w[4], w[5], w[6], w[7], w[8], w[9], w[10]); end proc; # Thomas Wieder, Nov 08 2007

MATHEMATICA

2^(2^Range[0, 9] - 1) - 1 (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)

PROG

(PARI) a(n)=if(n<1, 0, -1+2*(1+a(n-1))^2)

(Sage) [stirling_number2(2^(n-1), 2) for n in xrange(1, 10)] # Zerinvary Lajos, Nov 27 2009

CROSSREFS

Cf. A077586.

Sequence in context: A034670 A020516 A253851 * A261487 A134722 A053713

Adjacent sequences:  A077582 A077583 A077584 * A077586 A077587 A077588

KEYWORD

nonn

AUTHOR

Henry Bottomley, Nov 07 2002

EXTENSIONS

Corrected by Lekraj Beedassy, Jan 02 2007

STATUS

approved

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Last modified November 20 21:05 EST 2018. Contains 317422 sequences. (Running on oeis4.)