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A320171
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Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
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5
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1, 2, 5, 11, 29, 82, 247, 782, 2579, 8702, 29975, 104818, 371111, 1327307, 4788687, 17404838, 63669763, 234237605, 866090021, 3216738344, 11995470691, 44894977263, 168582174353, 634939697164, 2398004674911, 9079614633247, 34458722286825, 131059771522401
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OFFSET
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1,2
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COMMENTS
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A rooted tree is series-reduced if every non-leaf node has at least two branches.
In an identity tree, all branches directly under any given node are different.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(4) = 11 rooted identity trees:
(1) (2) (3) (4)
(11) (21) (22)
(111) (31)
((1)(2)) (211)
((1)(11)) (1111)
((1)(3))
((1)(21))
((2)(11))
((1)(111))
((1)((1)(2)))
((1)((1)(11)))
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn], UnsameQ@@#&]], {mtn, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[gig[y]], {y, IntegerPartitions[n]}], {n, 8}]
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PROG
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(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numbpart(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018
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CROSSREFS
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Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A319312, A320172, A320177, A320178.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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