A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

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

Offset: 1

Author

Gus Wiseman, Jan 20 2020

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

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.

Programs

Mathematica

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]

Formula

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

Extensions

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

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