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A330463
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Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.
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8
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1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0
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OFFSET
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0,5
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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 2 0
0 3 2 0
0 5 4 0 0
0 7 11 1 0 0
0 11 20 6 0 0 0
0 15 40 16 0 0 0 0
0 22 68 40 3 0 0 0 0
...
Row n = 5 counts the following sets of multisets:
{{5}} {{1},{4}} {{1},{2},{1,1}}
{{1,4}} {{2},{3}}
{{2,3}} {{1},{1,3}}
{{1,1,3}} {{1},{2,2}}
{{1,2,2}} {{2},{1,2}}
{{1,1,1,2}} {{3},{1,1}}
{{1,1,1,1,1}} {{1},{1,1,2}}
{{1,1},{1,2}}
{{2},{1,1,1}}
{{1},{1,1,1,1}}
{{1,1},{1,1,1}}
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
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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@@#, Length[#]==k]&]], {n, 0, 10}, {k, 0, n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
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PROG
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(PARI)
A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(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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KEYWORD
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AUTHOR
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STATUS
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approved
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