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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.
26
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, 56, 56, 74, 74, 74, 56, 56, 32, 32, 64, 64, 112, 112, 148, 148, 175, 148, 148, 112, 112, 64, 64, 128, 128, 224, 224, 296, 296, 350, 350, 350, 296, 296, 224, 224, 128, 128
OFFSET
2,2
COMMENTS
Rows are palindromic.
EXAMPLE
Triangle begins:
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 56 56 74 74 74 56 56 32 32
Row n = 4 counts the following subsets:
{1,2} {1,3} {1,4} {2,4} {3,4}
{1,2,3} {1,2,3} {2,3} {1,2,4} {1,3,4}
{1,2,4} {1,3,4} {1,2,3} {2,3,4} {2,3,4}
{1,2,3,4} {1,2,3,4} {1,2,4} {1,2,3,4} {1,2,3,4}
{1,3,4}
{2,3,4}
{1,2,3,4}
MATHEMATICA
Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#, {2}], k]&]], {n, 2, 11}, {k, 3, 2n-1}]
CROSSREFS
Row lengths are A005408.
The case counting only length-2 subsets is A008967.
Column k = n + 1 appears to be A167762.
The version for all subsets (instead of just pairs) is A365381.
Column k = n is A365544.
A000009 counts subsets summing to n.
A007865/A085489/A151897 count certain types of sum-free subsets.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A093971/A088809/A364534 count certain types of sum-full subsets.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Sequence in context: A326544 A326683 A239961 * A308855 A301588 A308921
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Sep 15 2023
STATUS
approved