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A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

%I A358459 #6 Nov 19 2022 08:56:49
%S A358459 1,2,3,4,5,8,9,11,16,17,32,35,37,41,43,64,128,129,137,139,163,169,171,
%T A358459 256,257,293,512,515,529,547,553,555,641,649,651,675,681,683,1024,
%U A358459 1025,2048,2053,2057,2059,2177,2185,2187,2211,2217,2219,2305,2341,2563
%N A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).
%C A358459 An ordered tree is balanced if all leaves have the same distance from the root.
%C A358459 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 A358459 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 A358459 The terms together with their corresponding ordered trees begin:
%e A358459    1: o
%e A358459    2: (o)
%e A358459    3: ((o))
%e A358459    4: (oo)
%e A358459    5: (((o)))
%e A358459    8: (ooo)
%e A358459    9: ((oo))
%e A358459   11: ((o)(o))
%e A358459   16: (oooo)
%e A358459   17: ((((o))))
%e A358459   32: (ooooo)
%e A358459   35: ((oo)(o))
%e A358459   37: (((o))((o)))
%e A358459   41: ((o)(oo))
%e A358459   43: ((o)(o)(o))
%t A358459 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358459 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358459 Select[Range[100],SameQ@@Length/@Position[srt[#],{}]&]
%Y A358459 These trees are counted by A007059.
%Y A358459 The unordered version is A184155, counted by A048816.
%Y A358459 A000108 counts ordered rooted trees, unordered A000081.
%Y A358459 A358379 gives depth of standard ordered trees.
%Y A358459 Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.
%K A358459 nonn
%O A358459 1,2
%A A358459 _Gus Wiseman_, Nov 19 2022