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A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

%I A358376 #8 Nov 14 2022 20:00:52
%S A358376 1,4,8,16,18,25,32,36,50,57,64,72,100,114,121,128,137,144,200,228,242,
%T A358376 249,256,258,274,281,288,385,393,400,456,484,498,505,512,516,548,562,
%U A358376 569,576,770,786,793,800,897,905,912,968,996,1010,1017,1024,1032,1096
%N A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).
%C A358376 We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
%H A358376 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A358376 The initial terms and their corresponding trees:
%e A358376     1: o
%e A358376     4: (oo)
%e A358376     8: (ooo)
%e A358376    16: (oooo)
%e A358376    18: ((oo)o)
%e A358376    25: (o(oo))
%e A358376    32: (ooooo)
%e A358376    36: ((oo)oo)
%e A358376    50: (o(oo)o)
%e A358376    57: (oo(oo))
%e A358376    64: (oooooo)
%e A358376    72: ((oo)ooo)
%e A358376   100: (o(oo)oo)
%e A358376   114: (oo(oo)o)
%e A358376   121: (ooo(oo))
%e A358376   128: (ooooooo)
%e A358376   137: ((oo)(oo))
%e A358376   144: ((oo)oooo)
%e A358376   200: (o(oo)ooo)
%t A358376 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358376 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358376 Select[Range[100],FreeQ[srt[#],_[__]?(Length[#]==1&)]&]
%Y A358376 These trees are counted by A005043.
%Y A358376 The series-reduced case appears to be counted by A284778.
%Y A358376 The unordered version is A291636, counted by A001678.
%Y A358376 A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358376 A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y A358376 A358374 ranks ordered identity trees, counted by A032027.
%Y A358376 A358375 ranks ordered binary trees, counted by A126120.
%Y A358376 Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.
%K A358376 nonn
%O A358376 1,2
%A A358376 _Gus Wiseman_, Nov 14 2022