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A283261 Product of the different products of subsets of the set of numbers from 1 to n.

%I A283261 #31 Aug 01 2022 12:56:06
%S A283261 1,1,2,36,331776,42998169600000000,
%T A283261 13974055172471046820331520000000000000,
%U A283261 1833132881579690383668380351534446872452674453158326975200092938148249600000000000000000000000000
%N A283261 Product of the different products of subsets of the set of numbers from 1 to n.
%C A283261 Product of numbers in n-th row of A070861.
%H A283261 Alois P. Heinz, <a href="/A283261/b283261.txt">Table of n, a(n) for n = 0..10</a>
%F A283261 a(n) <= n!^((A000005(n!))/2) = n!^(A027423(n)/2). - _David A. Corneth_, Mar 05 2017
%F A283261 a(n) = n!^(A263292(n)). - _David A. Corneth_, Mar 06 2017
%e A283261 Rows with subsets of the sets of numbers from 1 to n:
%e A283261   {},
%e A283261   {}, {1};
%e A283261   {}, {1}, {2}, {1, 2};
%e A283261   {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3};
%e A283261   ...
%e A283261 Rows with the products of elements of these subsets:
%e A283261   1;
%e A283261   1, 1;
%e A283261   1, 1, 2, 2;
%e A283261   1, 1, 2, 3, 2, 3, 6, 6;
%e A283261   ...
%e A283261 Rows with the different products of elements of these subsets:
%e A283261   1;
%e A283261   1;
%e A283261   1, 2;
%e A283261   1, 2, 3, 6;
%e A283261   ...
%e A283261 a(0) = 1, a(1) = (1), a(2) = (1*2) = 2, a(3) = (1*2*3*6) = 36, ... .
%p A283261 b:= proc(n) option remember; `if`(n=0, {1},
%p A283261       map(x-> [x, x*n][], b(n-1)))
%p A283261     end:
%p A283261 a:= n-> mul(i, i=b(n)):
%p A283261 seq(a(n), n=0..7);  # _Alois P. Heinz_, Aug 01 2022
%t A283261 Table[Times @@ Union@ Map[Times @@ # &, Subsets@ Range@ n], {n, 7}] (* _Michael De Vlieger_, Mar 05 2017 *)
%o A283261 (PARI) a(n)=my(v=[2..n]); factorback(Set(vector(2^(n-1),i, factorback(vecextract(v,i-1))))) \\ _Charles R Greathouse IV_, Mar 06 2017
%Y A283261 Cf. A000005, A000142, A027423, A060957, A070863, A052295, A263292.
%K A283261 nonn
%O A283261 0,3
%A A283261 _Jaroslav Krizek_, Mar 04 2017
%E A283261 a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2022