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%I #10 Feb 28 2020 12:55:19
%S 1,1,1,4,6,11,26,42,78,148,280,481,867,1569,2742,4933,8493,14857,
%T 25925,44877,77022,132511,226449,385396,657314,1111115,1875708,
%U 3157379,5309439,8885889,14861478,24760339,41162971,68328959,113099231,186926116,308230044
%N Number of set partitions of set-systems with total sum n.
%C Number of sets of disjoint nonempty sets of nonempty sets of positive integers with total sum n.
%H Andrew Howroyd, <a href="/A330459/b330459.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_k A330462(n,k) * A000110(k).
%e The a(6) = 26 partitions:
%e ((6)) ((15)) ((123)) ((1)(2)(12))
%e ((24)) ((1)(14)) ((1))((2)(12))
%e ((1)(5)) ((1)(23)) ((12))((1)(2))
%e ((2)(4)) ((2)(13)) ((2))((1)(12))
%e ((1))((5)) ((3)(12)) ((1))((2))((12))
%e ((2))((4)) ((1))((14))
%e ((1))((23))
%e ((1)(2)(3))
%e ((2))((13))
%e ((3))((12))
%e ((1))((2)(3))
%e ((2))((1)(3))
%e ((3))((1)(2))
%e ((1))((2))((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,3],And[UnsameQ@@Join@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]
%o (PARI) \\ here L is A000009 and BellP is A000110 as series.
%o L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
%o BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
%o seq(n)={my(c=L(n), b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c, k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ _Andrew Howroyd_, Dec 29 2019
%Y Cf. A007713, A050342, A050343, A279375, A279785, A283877, A294617, A330460, A330462, A323787-A323795, A330452-A330459.
%K nonn
%O 0,4
%A _Gus Wiseman_, Dec 17 2019
%E Terms a(18) and beyond from _Andrew Howroyd_, Dec 29 2019
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