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A330462
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Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.
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9
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1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 0, 0, 0, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,8
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
0 1
0 1 0
0 2 1 0
0 2 2 0 0
0 3 4 0 0 0
0 4 6 2 0 0 0
0 5 11 3 0 0 0 0
0 6 16 8 0 0 0 0 0
0 8 25 15 1 0 0 0 0 0
0 10 35 28 4 0 0 0 0 0 0
...
Row n = 7 counts the following set-systems:
{{7}} {{1},{6}} {{1},{2},{4}}
{{1,6}} {{2},{5}} {{1},{2},{1,3}}
{{2,5}} {{3},{4}} {{1},{3},{1,2}}
{{3,4}} {{1},{1,5}}
{{1,2,4}} {{1},{2,4}}
{{2},{1,4}}
{{2},{2,3}}
{{3},{1,3}}
{{4},{1,2}}
{{1},{1,2,3}}
{{1,2},{1,3}}
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MATHEMATICA
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ppl[n_, k_]:=Switch[k, 0, {n}, 1, IntegerPartitions[n], _, Join@@Table[Union[Sort/@Tuples[ppl[#, k-1]&/@ptn]], {ptn, IntegerPartitions[n]}]];
Table[Length[Select[ppl[n, 2], And[UnsameQ@@#, And@@UnsameQ@@@#, Length[#]==k]&]], {n, 0, 10}, {k, 0, n}]
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PROG
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(PARI)
L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c, k)))); vector(#v, n, Vecrev(v[n], n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019
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CROSSREFS
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Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.
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KEYWORD
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AUTHOR
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STATUS
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approved
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