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

0, 0, 0, 4, 65, 1026, 17857, 349224, 7657281, 186895270, 5037424601, 148805552556, 4784793219505, 166458635341194, 6231891513395745, 249886992888096976, 10686839817678846209, 485632267141865950926, 23370062118676064101801, 1187393725239246382405140

Offset: 1

Author

Gus Wiseman, Jan 21 2020

Keywords

• nonn

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

The series-reduced version is A331578.

The unlabeled version is A331233.

Labeled rooted trees are counted by A000169.

Cf. A000081, A033185, A060313, A060356, A206429, A331488, A331490.

Programs

Mathematica

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}]

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

Formula

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