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A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

%I A326441 #19 Oct 09 2019 04:29:21
%S A326441 0,1,0,1,0,1,1,1,1,1,3,1,2,1,3,3,3,3,1,4,4,3,2,2,4,3,5,3,2,4,5,4,5,6,
%T A326441 1,4,2,5,4,7,4,4,3,3,6,14,3,4,10,6,3,6,4,4,4,8,7,6,8,7,10,5,11,8,5,11,
%U A326441 4,7,7,5,8,12,5,6,9,8,11,8,5,8,9,8,10,8
%N A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.
%C A326441 Essentially the same as A178830. - _R. J. Mathar_, Jul 12 2019
%H A326441 Giovanni Resta, <a href="/A326441/b326441.txt">Table of n, a(n) for n = 0..500</a>
%e A326441 The initial terms count the following subsets:
%e A326441    1: {1}
%e A326441    3: {1,2}
%e A326441    5: {3,5}
%e A326441    6: {3,4,5}
%e A326441    7: {2,4,5,7}
%e A326441    8: {2,4,5,6,7}
%e A326441    9: {2,3,5,6,7,9}
%e A326441   10: {4,5,6,8,9,10}
%e A326441   10: {2,3,5,6,7,8,9}
%e A326441   10: {1,2,3,4,5,8,9,10}
%e A326441 Also the number of subsets of {1..n} whose product is equal to the sum of their complement. For example, the initial terms count the following subsets:
%e A326441    1: {}
%e A326441    3: {3}
%e A326441    5: {1,2,4}
%e A326441    6: {1,2,6}
%e A326441    7: {1,3,6}
%e A326441    8: {1,3,8}
%e A326441    9: {1,4,8}
%e A326441   10: {6,7}
%e A326441   10: {1,4,10}
%e A326441   10: {1,2,3,7}
%p A326441 b:= proc(n, s, p)
%p A326441       `if`(s=p, 1, `if`(n<1, 0, b(n-1, s, p)+
%p A326441       `if`(s-n<p*n, 0, b(n-1, s-n, p*n))))
%p A326441     end:
%p A326441 a:= n-> b(n, n*(n+1)/2, 1):
%p A326441 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 12 2019
%t A326441 Table[Length[Select[Subsets[Range[n]],Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
%Y A326441 Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325044, A325538, A326172, A326173, A326174, A326175, A326179, A326180.
%K A326441 nonn
%O A326441 0,11
%A A326441 _Gus Wiseman_, Jul 07 2019
%E A326441 a(21)-a(83) from _Giovanni Resta_, Jul 08 2019