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A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

%I A371791 #8 Mar 21 2025 09:51:18
%S A371791 1,1,1,2,4,8,18,38,82,175,373,787,1651,3439,7126,14667,30049,61249,
%T A371791 124440,251922,508779,1025183,2062287,4142644,8312927,16667005,
%U A371791 33395275,66880828,133892910,267976571,536225921,1072842931,2146233971,4293248183,8587569636,17176654105,34355356676,68713584720,137430991937,274867311960,549741605972,1099492913172,2198998307679,4398013970156,8796049891377,17592130283755,35184298506429
%N A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.
%C A371791 A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
%e A371791 For S = {1,3,4,6} we have {{1,6},{3,4}}, so S is counted under a(6).
%e A371791 The a(0) = 1 through a(6) = 18 subsets:
%e A371791   {}  {}  {}  {}       {}         {}         {}
%e A371791               {1,2,3}  {1,2,3}    {1,2,3}    {1,2,3}
%e A371791                        {1,3,4}    {1,3,4}    {1,3,4}
%e A371791                        {1,2,3,4}  {1,4,5}    {1,4,5}
%e A371791                                   {2,3,5}    {1,5,6}
%e A371791                                   {1,2,3,4}  {2,3,5}
%e A371791                                   {1,2,4,5}  {2,4,6}
%e A371791                                   {2,3,4,5}  {1,2,3,4}
%e A371791                                              {1,2,3,6}
%e A371791                                              {1,2,4,5}
%e A371791                                              {1,2,5,6}
%e A371791                                              {1,3,4,6}
%e A371791                                              {2,3,4,5}
%e A371791                                              {2,3,5,6}
%e A371791                                              {3,4,5,6}
%e A371791                                              {1,2,3,4,6}
%e A371791                                              {1,2,4,5,6}
%e A371791                                              {2,3,4,5,6}
%t A371791 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371791 Table[Length[Select[Subsets[Range[n]],biqQ]],{n,0,15}]
%Y A371791 First differences are A232466.
%Y A371791 The complement is counted by A371792, differences A371793.
%Y A371791 This is the "bi-" case of A371796, differences A371797.
%Y A371791 A002219 aerated counts biquanimous partitions, ranks A357976.
%Y A371791 A006827 and A371795 count non-biquanimous partitions, ranks A371731.
%Y A371791 A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A371791 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371791 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371791 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371791 A321452 counts quanimous partitions, ranks A321454.
%Y A371791 A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371791 A371737 counts quanimous strict partitions, complement A371736.
%Y A371791 A371781 lists numbers with biquanimous prime signature, complement A371782.
%Y A371791 A371783 counts k-quanimous partitions.
%Y A371791 Cf. A035470, A064914, A357879, A365661, A366320, A365381, A365925, A367094, A371788, A371789.
%K A371791 nonn
%O A371791 0,4
%A A371791 _Gus Wiseman_, Apr 07 2024
%E A371791 a(16) onwards from _Martin Fuller_, Mar 21 2025