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

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 37, 43, 64, 128, 129, 137, 171, 256, 257, 293, 512, 529, 683, 1024, 1025, 2048, 2185, 2341, 2731, 4096, 8192, 10923, 16384, 16913, 18725, 32768, 32769, 32897, 34953, 43691, 65536, 65537, 131072, 131329, 149797, 174763

Offset: 1

Author

Gus Wiseman, Nov 14 2022

Keywords

• nonn

Comments

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.

A generalized Bethe tree is an unlabeled rooted tree where all branches directly under the same root are equal.

Examples

			The terms together with their corresponding ordered rooted trees begin:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    8: (ooo)
    9: ((oo))
   11: ((o)(o))
   16: (oooo)
   17: ((((o))))
   32: (ooooo)
   37: (((o))((o)))
   43: ((o)(o)(o))
   64: (oooooo)
  128: (ooooooo)
  129: ((ooo))
  137: ((oo)(oo))
  171: ((o)(o)(o)(o))
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

These trees are counted by A003238.

The unordered version is A214577, also counted by A003238.

A000081 counts unlabeled rooted trees, ranked by A358378.

A358371 and A358372 count leaves and nodes in standard ordered rooted trees.

A358374 ranks ordered identity trees, counted by A032027.

Cf. A001263, A004111, A004249, A005043, A063895, A276625, A331490, A358373, A358375, A358376.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(!SameQ@@#&)]&]