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A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

%I A325538 #19 Oct 09 2019 04:48:22
%S A325538 1,1,1,1,1,0,0,1,0,0,1,0,2,0,2,1,2,1,1,2,1,2,2,1,2,0,4,2,2,4,2,2,5,5,
%T A325538 1,2,2,3,2,5,3,4,2,2,3,10,2,4,7,5,3,3,7,6,4,4,5,5,5,2,6,4,6,5,3,8,4,5,
%U A325538 4,5,2,10,5,3,7,11,6,10,5,11,6,4,7,6,10
%N A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.
%C A325538 Also by definition the number of subsets whose sum is one fewer than the product of their complement.
%H A325538 Giovanni Resta, <a href="/A325538/b325538.txt">Table of n, a(n) for n = 0..2500</a>
%e A325538 The initial terms count the following subsets:
%e A325538    0: {}
%e A325538    1: {1}
%e A325538    2: {2}
%e A325538    3: {1,3}
%e A325538    4: {2,3}
%e A325538    7: {4,5}
%e A325538   10: {1,6,7}
%e A325538   12: {7,9}
%e A325538   12: {1,2,4,8}
%e A325538   14: {2,5,9}
%e A325538   14: {1,2,4,11}
%e A325538   15: {1,3,5,7}
%e A325538   16: {3,4,10}
%e A325538   16: {1,3,5,8}
%e A325538   17: {1,10,13}
%e A325538   18: {2,5,15}
%e A325538   19: {11,15}
%e A325538   19: {1,2,6,14}
%e A325538   20: {1,4,6,8}
%t A325538 Table[Length[Select[Subsets[Range[n]],1+Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
%t A325538 ric[n_, pr_, s_, lst_, t_] := Block[{k}, If[pr == t-s, cnt++]; Do[ If[pr k <= t, ric[n, pr k, s + k, k, t], Break[]], {k, lst+1, n}]]; a[n_] := (cnt = 0; ric[n, 1, 0, 0, n (n + 1)/2 + 1]; cnt); a /@ Range[0, 85] (* _Giovanni Resta_, Sep 13 2019 *)
%Y A325538 Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325041, A326172, A326173, A326174, A326175, A326179, A326180, A326441.
%K A325538 nonn
%O A325538 0,13
%A A325538 _Gus Wiseman_, Jul 07 2019
%E A325538 More terms from _Alois P. Heinz_, Jul 12 2019