cp's OEIS Frontend

A358377 Numbers k such that the k-th standard ordered rooted tree is a generalized Bethe tree (counted by A003238).

%I A358377 #5 Nov 14 2022 20:00:41
%S A358377 1,2,3,4,5,8,9,11,16,17,32,37,43,64,128,129,137,171,256,257,293,512,
%T A358377 529,683,1024,1025,2048,2185,2341,2731,4096,8192,10923,16384,16913,
%U A358377 18725,32768,32769,32897,34953,43691,65536,65537,131072,131329,149797,174763
%N A358377 Numbers k such that the k-th standard ordered rooted tree is a generalized Bethe tree (counted by A003238).
%C A358377 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 A358377 A generalized Bethe tree is an unlabeled rooted tree where all branches directly under the same root are equal.
%H A358377 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 A358377 The terms together with their corresponding ordered rooted trees begin:
%e A358377     1: o
%e A358377     2: (o)
%e A358377     3: ((o))
%e A358377     4: (oo)
%e A358377     5: (((o)))
%e A358377     8: (ooo)
%e A358377     9: ((oo))
%e A358377    11: ((o)(o))
%e A358377    16: (oooo)
%e A358377    17: ((((o))))
%e A358377    32: (ooooo)
%e A358377    37: (((o))((o)))
%e A358377    43: ((o)(o)(o))
%e A358377    64: (oooooo)
%e A358377   128: (ooooooo)
%e A358377   129: ((ooo))
%e A358377   137: ((oo)(oo))
%e A358377   171: ((o)(o)(o)(o))
%t A358377 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358377 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358377 Select[Range[1000],FreeQ[srt[#],_[__]?(!SameQ@@#&)]&]
%Y A358377 These trees are counted by A003238.
%Y A358377 The unordered version is A214577, also counted by A003238.
%Y A358377 A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358377 A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y A358377 A358374 ranks ordered identity trees, counted by A032027.
%Y A358377 Cf. A001263, A004111, A004249, A005043, A063895, A276625, A331490, A358373, A358375, A358376.
%K A358377 nonn
%O A358377 1,2
%A A358377 _Gus Wiseman_, Nov 14 2022