cp's OEIS Frontend

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

%I A358524 #6 Nov 21 2022 22:01:53
%S A358524 0,2,10,12,42,52,56,170,204,212,232,240,682,820,844,852,920,936,976,
%T A358524 992,2730,3276,3284,3380,3404,3412,3640,3688,3736,3752,3888,3920,4000,
%U A358524 4032,10922,13108,13132,13140,13516,13524,13620,13644,13652,14568,14744,14760
%N A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).
%C A358524 An ordered tree is balanced if all leaves are the same distance from the root.
%C A358524 The binary encoding of an ordered tree (see A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.
%e A358524 The terms together with their corresponding trees begin:
%e A358524     0: o
%e A358524     2: (o)
%e A358524    10: (oo)
%e A358524    12: ((o))
%e A358524    42: (ooo)
%e A358524    52: ((oo))
%e A358524    56: (((o)))
%e A358524   170: (oooo)
%e A358524   204: ((o)(o))
%e A358524   212: ((ooo))
%e A358524   232: (((oo)))
%e A358524   240: ((((o))))
%e A358524   682: (ooooo)
%e A358524   820: ((o)(oo))
%e A358524   844: ((oo)(o))
%e A358524   852: ((oooo))
%e A358524   920: (((o)(o)))
%e A358524   936: (((ooo)))
%e A358524   976: ((((oo))))
%e A358524   992: (((((o)))))
%t A358524 binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
%t A358524 bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
%t A358524 Select[Range[0,1000],binbalQ[#]&&SameQ@@Length/@Position[bint[#],{}]&]
%Y A358524 These trees are counted by A007059.
%Y A358524 This is a subset of A014486.
%Y A358524 The version for binary trees is A057122.
%Y A358524 The unordered version is A184155, counted by A048816.
%Y A358524 Another ranking of balanced ordered trees is A358459.
%Y A358524 A000108 counts ordered rooted trees, unordered A000081.
%Y A358524 A358453 counts transitive ordered trees, unordered A290689.
%Y A358524 Cf. A001263, A003238, A244925, A358377, A358379, A358523.
%K A358524 nonn
%O A358524 1,2
%A A358524 _Gus Wiseman_, Nov 21 2022