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 A077585 a(n) = 2^(2^n-1) - 1. 10

%I

%S 0,1,7,127,32767,2147483647,9223372036854775807,

%T 170141183460469231731687303715884105727,

%U 57896044618658097711785492504343953926634992332820282019728792003956564819967

%N a(n) = 2^(2^n-1) - 1.

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

%C 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, 2], ], [[1, 2], , ], [[1, 2], ], [, ], []}. - _Thomas Wieder_, Nov 08 2007

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleMersenneNumber.html">Double Mersenne Number</a>

%F a(n) = A000225(A000225(n)).

%F a(n) = A058891(n+1)-1. - corrected by _Maurizio De Leo_, Feb 25 2015

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

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

%F a(n) = sum((binomial(2^n-1,k)), k=1..2^n-1). - _Thomas Wieder_, Nov 08 2007

%F a(n) = 2 * a(n-1)^2 + 4 * a(n-1) + 1. - _Roderick MacPhee_, Oct 05 2012

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

%p 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,w,w,w,w,w,w,w,w,w); end proc; # _Thomas Wieder_, Nov 08 2007

%t 2^(2^Range[0, 9] - 1) - 1 (* _Vladimir Joseph Stephan Orlovsky_, Jun 22 2011 *)

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

%o (Sage) [stirling_number2(2^(n-1),2) for n in xrange(1,10)] # _Zerinvary Lajos_, Nov 27 2009

%Y Cf. A077586.

%K nonn

%O 0,3

%A _Henry Bottomley_, Nov 07 2002

%E Corrected by _Lekraj Beedassy_, Jan 02 2007

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Last modified November 17 18:24 EST 2019. Contains 329241 sequences. (Running on oeis4.)