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A365381
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Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.
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32
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1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 5, 2, 2, 1, 16, 8, 8, 10, 10, 7, 5, 5, 2, 2, 1, 32, 16, 16, 20, 20, 23, 15, 15, 12, 12, 8, 5, 5, 2, 2, 1, 64, 32, 32, 40, 40, 46, 47, 38, 33, 35, 29, 28, 21, 17, 14, 13, 8, 5, 5, 2, 2, 1, 128, 64, 64, 80, 80, 92, 94, 102, 79, 82, 76, 75, 68, 64, 53, 48, 43, 34, 33, 23, 19, 15, 13, 8, 5, 5, 2, 2, 1
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OFFSET
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0,2
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COMMENTS
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Row lengths are A000124(n) = 1 + n*(n+1)/2.
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LINKS
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EXAMPLE
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Triangle begins:
1
2 1
4 2 2 1
8 4 4 5 2 2 1
16 8 8 10 10 7 5 5 2 2 1
32 16 16 20 20 23 15 15 12 12 8 5 5 2 2 1
64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13 8 5 5 2 2 1
Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
-------------------------------------------------------
n=0: 1
n=1: 2 1
n=2: 4 2 2 1
n=3: 8 4 4 5 2 2 1
n=4: 16 8 8 10 10 7 5 5 2 2
n=5: 32 16 16 20 20 23 15 15 12 12
n=6: 64 32 32 40 40 46 47 38 33 35
n=7: 128 64 64 80 80 92 94 102 79 82
n=8: 256 128 128 160 160 184 188 204 207 184
n=9: 512 256 256 320 320 368 376 408 414 440
The T(5,8) = 12 subsets are:
{3,5} {1,2,5} {1,2,3,4} {1,2,3,4,5}
{1,3,4} {1,2,3,5}
{1,3,5} {1,2,4,5}
{2,3,5} {1,3,4,5}
{3,4,5} {2,3,4,5}
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MATHEMATICA
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Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 8}, {k, 0, n*(n+1)/2}]
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CROSSREFS
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Row lengths are A000124 = number of distinct sums of subsets of {1..n}.
Central column/main diagonal is A365376.
A000124 counts distinct possible sums of subsets of {1..n}.
Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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