cp's OEIS Frontend

A371793 Number of non-biquanimous subsets of {1..n} containing n.

%I A371793 #9 Mar 21 2025 09:59:08
%S A371793 1,2,3,6,12,22,44,84,163,314,610,1184,2308,4505,8843,17386,34336,
%T A371793 67881,134662,267431,532172,1060048,2113947,4218325,8423138,16826162,
%U A371793 33623311,67205646,134351795,268621562,537124814,1074092608,2147953084,4295613139,8590784715,17181035797,34361248692,68721546255,137441586921,274881519876,549760320576,1099517861045,2199030848627,4398057100987,8796105652038,17592203866158
%N A371793 Number of non-biquanimous subsets of {1..n} containing n.
%C A371793 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 A371793 The a(1) = 1 through a(5) = 12 subsets:
%e A371793   {1}  {2}    {3}    {4}      {5}
%e A371793        {1,2}  {1,3}  {1,4}    {1,5}
%e A371793               {2,3}  {2,4}    {2,5}
%e A371793                      {3,4}    {3,5}
%e A371793                      {1,2,4}  {4,5}
%e A371793                      {2,3,4}  {1,2,5}
%e A371793                               {1,3,5}
%e A371793                               {2,4,5}
%e A371793                               {3,4,5}
%e A371793                               {1,2,3,5}
%e A371793                               {1,3,4,5}
%e A371793                               {1,2,3,4,5}
%t A371793 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371793 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!biqQ[#]&]],{n,15}]
%Y A371793 The complement is counted by A232466, differences of A371791.
%Y A371793 This is the "bi-" version of A371790, differences of A371789.
%Y A371793 First differences of A371792.
%Y A371793 The complement is the "bi-" version of A371797, differences of A371796.
%Y A371793 A002219 aerated counts biquanimous partitions, ranks A357976.
%Y A371793 A006827 and A371795 count non-biquanimous partitions, ranks A371731.
%Y A371793 A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A371793 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371793 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371793 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371793 A321452 counts quanimous partitions, ranks A321454.
%Y A371793 A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371793 A371737 counts quanimous strict partitions, complement A371736.
%Y A371793 A371781 lists numbers with biquanimous prime signature, complement A371782.
%Y A371793 A371783 counts k-quanimous partitions.
%Y A371793 Cf. A035470, A064914, A365543, A365661, A365663, A366320, A365381, A367094, A371788.
%K A371793 nonn
%O A371793 1,2
%A A371793 _Gus Wiseman_, Apr 07 2024
%E A371793 a(16) onwards from _Martin Fuller_, Mar 21 2025