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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.
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%I #9 Sep 17 2023 12:08:03

%S 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 56,56,74,74,74,56,56,32,32,64,64,112,112,148,148,175,148,148,112,112,

%U 64,64,128,128,224,224,296,296,350,350,350,296,296,224,224,128,128

%N 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 Rows are palindromic.

%e Triangle begins:

%e 1

%e 2 2 2

%e 4 4 7 4 4

%e 8 8 14 14 14 8 8

%e 16 16 28 28 37 28 28 16 16

%e 32 32 56 56 74 74 74 56 56 32 32

%e Row n = 4 counts the following subsets:

%e {1,2} {1,3} {1,4} {2,4} {3,4}

%e {1,2,3} {1,2,3} {2,3} {1,2,4} {1,3,4}

%e {1,2,4} {1,3,4} {1,2,3} {2,3,4} {2,3,4}

%e {1,2,3,4} {1,2,3,4} {1,2,4} {1,2,3,4} {1,2,3,4}

%e {1,3,4}

%e {2,3,4}

%e {1,2,3,4}

%t Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#,{2}],k]&]], {n,2,11}, {k,3,2n-1}]

%Y Row lengths are A005408.

%Y The case counting only length-2 subsets is A008967.

%Y Column k = n + 1 appears to be A167762.

%Y The version for all subsets (instead of just pairs) is A365381.

%Y Column k = n is A365544.

%Y A000009 counts subsets summing to n.

%Y A007865/A085489/A151897 count certain types of sum-free subsets.

%Y A046663 counts partitions with no submultiset summing to k, strict A365663.

%Y A093971/A088809/A364534 count certain types of sum-full subsets.

%Y A365543 counts partitions with a submultiset summing to k, strict A365661.

%Y Cf. A068911, A095944, A238628, A288728, A326083, A364272, A365376, A365377.

%K nonn,tabf

%O 2,2

%A _Gus Wiseman_, Sep 15 2023