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A331577
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Number of labeled rooted trees with n vertices and more than two branches of the root.
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3
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0, 0, 0, 4, 65, 1026, 17857, 349224, 7657281, 186895270, 5037424601, 148805552556, 4784793219505, 166458635341194, 6231891513395745, 249886992888096976, 10686839817678846209, 485632267141865950926, 23370062118676064101801, 1187393725239246382405140
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OFFSET
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1,4
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LINKS
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FORMULA
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For n > 1, a(n) = Sum_{k > 2} A206429(n, k).
E.g.f.: f(x) - x*(1 + f(x) + f(x)^2/2), where f(x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 23 2020
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EXAMPLE
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Non-isomorphic representatives of the a(6) = 1026 trees (in the format root[branches]) are:
1[2,3,4[5[6]]]
1[2,3[4],5[6]]
1[2,3,4[5,6]]
1[2,3,4,5[6]]
1[2,3,4,5,6]
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MATHEMATICA
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lrt[set_]:=If[Length[set]==0, {}, Join@@Table[Apply[root, #]&/@Join@@Table[Tuples[lrt/@stn], {stn, sps[DeleteCases[set, root]]}], {root, set}]];
Table[Length[Select[lrt[Range[n]], Length[#]>2&]], {n, 6}]
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PROG
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(PARI) seq(n)={my(f=serreverse(x*exp(O(x^n) -x ))); Vec(serlaplace(f - x*(1 + f + f^2/2)), -n)} \\ Andrew Howroyd, Jan 23 2020
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CROSSREFS
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The series-reduced version is A331578.
Labeled rooted trees are counted by A000169.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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