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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.

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