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A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root). 9
0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 70, 134, 263, 513, 1022, 2030, 4076, 8203, 16614, 33738, 68833, 140796, 288989, 594621, 1226781, 2536532, 5256303, 10913196, 22700682, 47299699, 98714362, 206323140, 431847121, 905074333, 1899247187, 3990145833, 8392281473 (list; graph; refs; listen; history; text; internal format)



Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.


Table of n, a(n) for n=1..38.

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)

Eric Weisstein's World of Mathematics, Series-reduced tree.

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.


For n > 1, a(n) = A001679(n) - A001678(n).


The a(4) = 1 through a(9) = 10 trees:

(ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)

(oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))

(ooo(oo)) (ooo(ooo)) (ooo(oooo))

(oooo(oo)) (oooo(ooo))

(o(oo)(oo)) (ooooo(oo))

(oo(o(oo))) (o(oo)(ooo))






urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]], {ptn, IntegerPartitions[n-1]}];

Table[Length[Select[urt[n], Length[#]>2&&FreeQ[#, {_}]&]], {n, 10}]


The not necessarily lone-child-avoiding version is A331233.

The Matula-Goebel numbers of these trees are listed by A331490.

A000081 counts unlabeled rooted trees.

A001678 counts lone-child-avoiding rooted trees.

A001679 counts topologically series-reduced rooted trees.

A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.

A331489 lists Matula-Goebel numbers of series-reduced rooted trees.

Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

Sequence in context: A047131 A231331 A008927 * A052525 A006606 A120421

Adjacent sequences: A331485 A331486 A331487 * A331489 A331490 A331491




Gus Wiseman, Jan 20 2020


a(37)-a(38) from Jinyuan Wang, Jun 26 2020

Terminology corrected (lone-child-avoiding, not series-reduced) by Gus Wiseman, May 10 2021



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Last modified December 9 10:28 EST 2022. Contains 358700 sequences. (Running on oeis4.)