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%I #16 Jan 24 2020 11:45:11
%S 0,0,0,4,65,1026,17857,349224,7657281,186895270,5037424601,
%T 148805552556,4784793219505,166458635341194,6231891513395745,
%U 249886992888096976,10686839817678846209,485632267141865950926,23370062118676064101801,1187393725239246382405140
%N Number of labeled rooted trees with n vertices and more than two branches of the root.
%H Andrew Howroyd, <a href="/A331577/b331577.txt">Table of n, a(n) for n = 1..200</a>
%F For n > 1, a(n) = Sum_{k > 2} A206429(n, k).
%F 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
%e Non-isomorphic representatives of the a(6) = 1026 trees (in the format root[branches]) are:
%e 1[2,3,4[5[6]]]
%e 1[2,3[4],5[6]]
%e 1[2,3,4[5,6]]
%e 1[2,3,4,5[6]]
%e 1[2,3,4,5,6]
%t lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
%t Table[Length[Select[lrt[Range[n]],Length[#]>2&]],{n,6}]
%o (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
%Y The series-reduced version is A331578.
%Y The unlabeled version is A331233.
%Y Labeled rooted trees are counted by A000169.
%Y Cf. A000081, A033185, A060313, A060356, A206429, A331488, A331490.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 21 2020
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