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A358375 Numbers k such that the k-th standard ordered rooted tree is binary.

%I A358375 #6 Nov 14 2022 16:06:42
%S A358375 1,4,18,25,137,262146,393217,2097161,2228225
%N A358375 Numbers k such that the k-th standard ordered rooted tree is binary.
%C A358375 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 A358375 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 A358375 The initial terms and their corresponding trees:
%e A358375        1: o
%e A358375        4: (oo)
%e A358375       18: ((oo)o)
%e A358375       25: (o(oo))
%e A358375      137: ((oo)(oo))
%e A358375   262146: (((oo)o)o)
%e A358375   393217: (o((oo)o))
%t A358375 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358375 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358375 Select[Range[1000],FreeQ[srt[#],_[__]?(Length[#]!=2&)]&]
%Y A358375 The unordered version is A111299, counted by A001190
%Y A358375 These trees are counted by A126120.
%Y A358375 A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358375 A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y A358375 Cf. A001263, A004249, A005043, A063895, A245824, A284778, A358373, A358374, A358376, A358377.
%K A358375 nonn,more
%O A358375 1,2
%A A358375 _Gus Wiseman_, Nov 14 2022