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%I #9 Oct 25 2018 22:21:00
%S 1,2,4,8,19,53,151,459,1445,4634,15154,50253,168607,571212,1951588,
%T 6715575,23255444,80978697,283373024,995995996,3514614634,12446666967,
%U 44222390525,157587392768,563096832839,2017121728223,7242436444030,26059512879605,93952946906117
%N Number of series-reduced rooted identity 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.
%C In an identity tree, all branches directly under any given node are different.
%H Andrew Howroyd, <a href="/A320178/b320178.txt">Table of n, a(n) for n = 1..200</a>
%e The a(1) = 1 through a(5) = 19 rooted trees:
%e (1) (2) (3) (4) (5)
%e (11) (111) (22) (11111)
%e ((1)(2)) (1111) ((1)(4))
%e ((1)(11)) ((1)(3)) ((2)(3))
%e ((2)(11)) ((1)(22))
%e ((1)(111)) ((3)(11))
%e ((1)((1)(2))) ((2)(111))
%e ((1)((1)(11))) ((1)(1111))
%e ((11)(111))
%e ((1)(2)(11))
%e ((1)((1)(3)))
%e ((2)((1)(2)))
%e ((11)((1)(2)))
%e ((1)((2)(11)))
%e ((2)((1)(11)))
%e ((1)((1)(111)))
%e ((11)((1)(11)))
%e ((1)((1)((1)(2))))
%e ((1)((1)((1)(11))))
%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 gob[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gob/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
%t Table[Length[Join@@Table[gob[m],{m,IntegerPartitions[n]}]],{n,10}]
%o (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + WeighT(v[1..n])[n]); v} \\ _Andrew Howroyd_, Oct 25 2018
%Y Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.
%Y Cf. A320171, A320174, A320175, A320176, A320177.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 07 2018
%E Terms a(13) and beyond from _Andrew Howroyd_, Oct 25 2018
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