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%I #5 Nov 14 2022 20:00:41
%S 1,2,3,4,5,8,9,11,16,17,32,37,43,64,128,129,137,171,256,257,293,512,
%T 529,683,1024,1025,2048,2185,2341,2731,4096,8192,10923,16384,16913,
%U 18725,32768,32769,32897,34953,43691,65536,65537,131072,131329,149797,174763
%N Numbers k such that the k-th standard ordered rooted tree is a generalized Bethe tree (counted by A003238).
%C 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.
%C A generalized Bethe tree is an unlabeled rooted tree where all branches directly under the same root are equal.
%H 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 The terms together with their corresponding ordered rooted trees begin:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 4: (oo)
%e 5: (((o)))
%e 8: (ooo)
%e 9: ((oo))
%e 11: ((o)(o))
%e 16: (oooo)
%e 17: ((((o))))
%e 32: (ooooo)
%e 37: (((o))((o)))
%e 43: ((o)(o)(o))
%e 64: (oooooo)
%e 128: (ooooooo)
%e 129: ((ooo))
%e 137: ((oo)(oo))
%e 171: ((o)(o)(o)(o))
%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t Select[Range[1000],FreeQ[srt[#],_[__]?(!SameQ@@#&)]&]
%Y These trees are counted by A003238.
%Y The unordered version is A214577, also counted by A003238.
%Y A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y A358374 ranks ordered identity trees, counted by A032027.
%Y Cf. A001263, A004111, A004249, A005043, A063895, A276625, A331490, A358373, A358375, A358376.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 14 2022
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