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A331577 Number of labeled rooted trees with n vertices and more than two branches of the root.

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