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

1,4

Links

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

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

Example

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]

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}]

Prog

(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

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.

Sequence in context: A041119 A278547 A293562 * A307169 A335176 A307185

Adjacent sequences: A331574 A331575 A331576 * A331578 A331579 A331580

Keyword

nonn

Author

Gus Wiseman, Jan 21 2020

Status

approved