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Number of collections F of subsets of {1,2,...,n} whose union is itself an element of F.

2

`%I #12 Aug 28 2014 20:08:09
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`%S 1,3,13,159,33337,2147648859,9223372049740171909,
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`%T 170141183460469231796250908018965844535,
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`%U 57896044618658097711785492504343953927996121800504035873840544850835832773873
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`%N Number of collections F of subsets of {1,2,...,n} whose union is itself an element of F.
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`%C Equivalently, the number of partial orders (on some subset of the powerset of {1,2,...,n} ordered by set inclusion) that contain a greatest element.
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`%H Alois P. Heinz, <a href="/A246418/b246418.txt">Table of n, a(n) for n = 0..11</a>
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`%F a(n) = Sum_{k=0..n} C(n,k)*2^(2^k-1).
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`%F a(n) = 2^(2^n) - A246537(n).
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`%e a(2) = 13 because there are 16 families of subsets of {1,2}. All of these contain their union except: {}, {{1},{2}}, {{},{1},{2}}. 16-3=13.
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`%t nn = 9; Table[Sum[Binomial[n, i] 2^(2^i - 1), {i, 0, n}], {n, 0, nn}]
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`%o (PARI) a(n)=sum(k=0,n, binomial(n,k)*2^(2^k-1));
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`%Y Cf. A246537.
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`%K nonn
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`%O 0,2
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`%A _Geoffrey Critzer_, Aug 25 2014
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