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Number of double-closed subsets of {1..n}.
19

%I #17 Jun 11 2019 00:28:31

%S 1,2,3,6,8,16,24,48,60,120,180,360,480,960,1440,2880,3456,6912,10368,

%T 20736,27648,55296,82944,165888,207360,414720,622080,1244160,1658880,

%U 3317760,4976640,9953280,11612160,23224320,34836480,69672960,92897280

%N Number of double-closed subsets of {1..n}.

%C These are subsets containing twice any element whose double is <= n.

%C Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:

%C {} {1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5} {1,2,3,4,5,6}

%C {3} {1,3} {1,2,4} {1,2,3,5} {1,2,3,4,6}

%C {5} {1,5} {1,2,5} {1,2,3,6} {1,2,3,5,6}

%C {3,5} {1,3,5} {1,2,4,5}

%C {3,6} {1,3,6} {1,3,5,6}

%C {3,5,6}

%H Charlie Neder, <a href="/A308546/b308546.txt">Table of n, a(n) for n = 0..500</a>

%F From _Charlie Neder_, Jun 10 2019: (Start)

%F a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).

%F a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

%e The a(6) = 24 subsets:

%e {} {4} {2,4} {1,2,4} {1,2,4,5} {1,2,3,4,6} {1,2,3,4,5,6}

%e {5} {3,6} {2,4,5} {1,2,4,6} {1,2,4,5,6}

%e {6} {4,5} {2,4,6} {2,3,4,6} {2,3,4,5,6}

%e {4,6} {3,4,6} {2,4,5,6}

%e {5,6} {3,5,6} {3,4,5,6}

%e {4,5,6}

%t Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[2*#,#<=n&]]&]],{n,0,10}]

%Y Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 06 2019

%E a(21)-a(36) from _Charlie Neder_, Jun 10 2019