cp's OEIS Frontend

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

%I A384350 #8 Jun 07 2025 16:45:56
%S A384350 0,0,0,1,4,13,33,81,183,402,856,1801,3721,7646,15567,31575
%N A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.
%C A384350 Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.
%e A384350 For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
%e A384350 The a(0) = 0 through a(5) = 13 subsets:
%e A384350   .  .  .  {3}  {3}    {3}
%e A384350                 {4}    {4}
%e A384350                 {2,4}  {5}
%e A384350                 {3,4}  {1,5}
%e A384350                        {2,4}
%e A384350                        {2,5}
%e A384350                        {3,4}
%e A384350                        {3,5}
%e A384350                        {4,5}
%e A384350                        {1,4,5}
%e A384350                        {2,3,5}
%e A384350                        {2,4,5}
%e A384350                        {3,4,5}
%t A384350 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384350 Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]
%Y A384350 The complement is counted by A326080, allowing repeats A326083.
%Y A384350 For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
%Y A384350 First differences are A384391.
%Y A384350 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384350 A179009 counts maximally refined strict partitions, ranks A383707.
%Y A384350 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384350 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384350 A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
%Y A384350 Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.
%K A384350 nonn,more
%O A384350 0,5
%A A384350 _Gus Wiseman_, Jun 05 2025