cp's OEIS Frontend

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

%I A331488 #21 Feb 16 2025 08:33:59
%S A331488 0,0,0,1,1,2,3,6,10,20,36,70,134,263,513,1022,2030,4076,8203,16614,
%T A331488 33738,68833,140796,288989,594621,1226781,2536532,5256303,10913196,
%U A331488 22700682,47299699,98714362,206323140,431847121,905074333,1899247187,3990145833,8392281473
%N A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).
%C A331488 Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.
%H A331488 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014)
%H A331488 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Series-ReducedTree.html">Series-reduced tree.</a>
%H A331488 Gus Wiseman, <a href="https://oeis.org/A001678/a001678.txt">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%F A331488 For n > 1, a(n) = A001679(n) - A001678(n).
%e A331488 The a(4) = 1 through a(9) = 10 trees:
%e A331488   (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)    (oooooooo)
%e A331488                  (oo(oo))  (oo(ooo))  (oo(oooo))   (oo(ooooo))
%e A331488                            (ooo(oo))  (ooo(ooo))   (ooo(oooo))
%e A331488                                       (oooo(oo))   (oooo(ooo))
%e A331488                                       (o(oo)(oo))  (ooooo(oo))
%e A331488                                       (oo(o(oo)))  (o(oo)(ooo))
%e A331488                                                    (oo(o(ooo)))
%e A331488                                                    (oo(oo)(oo))
%e A331488                                                    (oo(oo(oo)))
%e A331488                                                    (ooo(o(oo)))
%t A331488 urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
%t A331488 Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]
%Y A331488 The not necessarily lone-child-avoiding version is A331233.
%Y A331488 The Matula-Goebel numbers of these trees are listed by A331490.
%Y A331488 A000081 counts unlabeled rooted trees.
%Y A331488 A001678 counts lone-child-avoiding rooted trees.
%Y A331488 A001679 counts topologically series-reduced rooted trees.
%Y A331488 A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
%Y A331488 A331489 lists Matula-Goebel numbers of series-reduced rooted trees.
%Y A331488 Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.
%K A331488 nonn
%O A331488 1,6
%A A331488 _Gus Wiseman_, Jan 20 2020
%E A331488 a(37)-a(38) from _Jinyuan Wang_, Jun 26 2020
%E A331488 Terminology corrected (lone-child-avoiding, not series-reduced) by _Gus Wiseman_, May 10 2021