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%I #8 Oct 25 2018 22:19:15
%S 1,3,6,19,55,200,713,2740,10651,42637,173012,713280,2972389,12514188,
%T 53119400,227140464,977382586,4229274235,18391269922,80330516578,
%U 352269725526,1550357247476,6845517553493,30316222112019,134626183784975,599341552234773,2674393679352974
%N Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.
%C A rooted tree is series-reduced if every non-leaf node has at least two branches.
%H Andrew Howroyd, <a href="/A320174/b320174.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 1 through a(4) = 19 trees:
%e (1) (2) (3) (4)
%e (11) (111) (22)
%e ((1)(1)) ((1)(2)) (1111)
%e ((1)(11)) ((1)(3))
%e ((1)(1)(1)) ((2)(2))
%e ((1)((1)(1))) ((2)(11))
%e ((1)(111))
%e ((11)(11))
%e ((1)(1)(2))
%e ((1)(1)(11))
%e ((1)((1)(2)))
%e ((2)((1)(1)))
%e ((1)((1)(11)))
%e ((1)(1)(1)(1))
%e ((11)((1)(1)))
%e ((1)((1)(1)(1)))
%e ((1)(1)((1)(1)))
%e (((1)(1))((1)(1)))
%e ((1)((1)((1)(1))))
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t dot[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[dot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
%t Table[Length[Join@@Table[dot[m],{m,IntegerPartitions[n]}]],{n,10}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + EulerT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320175.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Oct 25 2018
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