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%I #17 Nov 24 2023 12:38:05
%S 0,0,1,2,7,15,38,79,184,378,823,1682,3552,7208,14948,30154,61698,
%T 124302,252125,506521,1022768,2051555,4127633,8272147,16607469,
%U 33258510,66680774,133467385,267349211,535007304,1071020315,2142778192,4288207796
%N Number of subsets of {1..n} with a subset summing to n + 1.
%F Diagonal k = n + 1 of A365381.
%e The subset S = {1,2,4} has subset {1,4} with sum 4+1 and {2,4} with sum 5+1 and {1,2,4} with sum 6+1, so S is counted under a(4), a(5), and a(6).
%e The a(0) = 0 through a(5) = 15 subsets:
%e . . {1,2} {1,3} {1,4} {1,5}
%e {1,2,3} {2,3} {2,4}
%e {1,2,3} {1,2,3}
%e {1,2,4} {1,2,4}
%e {1,3,4} {1,2,5}
%e {2,3,4} {1,3,5}
%e {1,2,3,4} {1,4,5}
%e {2,3,4}
%e {2,4,5}
%e {1,2,3,4}
%e {1,2,3,5}
%e {1,2,4,5}
%e {1,3,4,5}
%e {2,3,4,5}
%e {1,2,3,4,5}
%t Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],n+1]&]],{n,0,10}]
%o (Python)
%o from itertools import combinations
%o from sympy.utilities.iterables import partitions
%o def A366130(n):
%o a = tuple(set(p.keys()) for p in partitions(n+1,k=n) if max(p.values(),default=0)==1)
%o return sum(1 for k in range(2,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any(s<=w for s in a)) # _Chai Wah Wu_, Nov 24 2023
%Y For pairs summing to n + 1 we have A167762, complement A038754.
%Y For n instead of n + 1 we have A365376, for pairs summing to n A365544.
%Y The complement is counted by A365377 shifted.
%Y The complement for pairs summing to n is counted by A365377.
%Y A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
%Y A093971/A088809/A364534 count certain types of sum-full subsets.
%Y Cf. A004526, A004737, A008967, A046663, A238628, A365381, A365541.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Oct 07 2023
%E a(20)-a(32) from _Chai Wah Wu_, Nov 24 2023
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