cp's OEIS Frontend

A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

%I A365541 #9 Sep 17 2023 12:08:03
%S A365541 1,2,2,2,4,4,7,4,4,8,8,14,14,14,8,8,16,16,28,28,37,28,28,16,16,32,32,
%T A365541 56,56,74,74,74,56,56,32,32,64,64,112,112,148,148,175,148,148,112,112,
%U A365541 64,64,128,128,224,224,296,296,350,350,350,296,296,224,224,128,128
%N A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.
%C A365541 Rows are palindromic.
%e A365541 Triangle begins:
%e A365541     1
%e A365541     2    2    2
%e A365541     4    4    7    4    4
%e A365541     8    8   14   14   14    8    8
%e A365541    16   16   28   28   37   28   28   16   16
%e A365541    32   32   56   56   74   74   74   56   56   32   32
%e A365541 Row n = 4 counts the following subsets:
%e A365541   {1,2}      {1,3}      {1,4}      {2,4}      {3,4}
%e A365541   {1,2,3}    {1,2,3}    {2,3}      {1,2,4}    {1,3,4}
%e A365541   {1,2,4}    {1,3,4}    {1,2,3}    {2,3,4}    {2,3,4}
%e A365541   {1,2,3,4}  {1,2,3,4}  {1,2,4}    {1,2,3,4}  {1,2,3,4}
%e A365541                         {1,3,4}
%e A365541                         {2,3,4}
%e A365541                         {1,2,3,4}
%t A365541 Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#,{2}],k]&]], {n,2,11}, {k,3,2n-1}]
%Y A365541 Row lengths are A005408.
%Y A365541 The case counting only length-2 subsets is A008967.
%Y A365541 Column k = n + 1 appears to be A167762.
%Y A365541 The version for all subsets (instead of just pairs) is A365381.
%Y A365541 Column k = n is A365544.
%Y A365541 A000009 counts subsets summing to n.
%Y A365541 A007865/A085489/A151897 count certain types of sum-free subsets.
%Y A365541 A046663 counts partitions with no submultiset summing to k, strict A365663.
%Y A365541 A093971/A088809/A364534 count certain types of sum-full subsets.
%Y A365541 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365541 Cf. A068911, A095944, A238628, A288728, A326083, A364272, A365376, A365377.
%K A365541 nonn,tabf
%O A365541 2,2
%A A365541 _Gus Wiseman_, Sep 15 2023