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%I #13 Feb 22 2023 10:11:07
%S 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,
%T 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,
%U 0,0,12,52,46,9,0,0,0,0,0,0,0
%N 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.
%H Andrew Howroyd, <a href="/A330462/b330462.txt">Table of n, a(n) for n = 0..1325</a> (rows n = 0..50)
%F G.f.: Product_{j>=1} (1 + y*x^j)^A000009(j). - _Andrew Howroyd_, Dec 29 2019
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 0
%e 0 2 1 0
%e 0 2 2 0 0
%e 0 3 4 0 0 0
%e 0 4 6 2 0 0 0
%e 0 5 11 3 0 0 0 0
%e 0 6 16 8 0 0 0 0 0
%e 0 8 25 15 1 0 0 0 0 0
%e 0 10 35 28 4 0 0 0 0 0 0
%e ...
%e Row n = 7 counts the following set-systems:
%e {{7}} {{1},{6}} {{1},{2},{4}}
%e {{1,6}} {{2},{5}} {{1},{2},{1,3}}
%e {{2,5}} {{3},{4}} {{1},{3},{1,2}}
%e {{3,4}} {{1},{1,5}}
%e {{1,2,4}} {{1},{2,4}}
%e {{2},{1,4}}
%e {{2},{2,3}}
%e {{3},{1,3}}
%e {{4},{1,2}}
%e {{1},{1,2,3}}
%e {{1,2},{1,3}}
%t ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t Table[Length[Select[ppl[n,2],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
%o (PARI)
%o L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
%o 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))}
%o {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y Row sums are A050342.
%Y Column k = 1 is A000009.
%Y Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.
%K nonn,tabl
%O 0,8
%A _Gus Wiseman_, Dec 18 2019
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