A330462 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.

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

Offset: 0

Author

Gus Wiseman, Dec 18 2019

Keywords

• nonn
• tabl

Examples

			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}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Crossrefs

Row sums are A050342.

Column k = 1 is A000009.

Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.

Programs

Mathematica

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}]

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

Formula

G.f.: Product_{j>=1} (1 + y*x^j)^A000009(j). - Andrew Howroyd, Dec 29 2019