A216648 - cp's OEIS frontend

A216648 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string without totally balanced proper prefixes such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n).

%I A216648 #19 Jan 10 2015 11:09:31
%S A216648 2,12,52,56,212,216,232,240,852,856,872,880,920,936,944,976,992,3412,
%T A216648 3416,3432,3440,3480,3496,3504,3536,3552,3688,3696,3752,3760,3792,
%U A216648 3808,3888,3920,3936,4000,4032,13652,13656,13672,13680,13720,13736,13744,13776
%N A216648 Triangle T(n,k) in which n-th row lists in increasing order all positive integers with a representation as totally balanced 2n digit binary string without totally balanced proper prefixes such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n).
%C A216648 There is a simple bijection between the elements of row n and the rooted trees with n nodes.  Each matching pair (1,0) in the binary string representation encodes a node, each totally balanced substring encodes a list of subtrees.
%H A216648 Alois P. Heinz, <a href="/A216648/b216648.txt">Rows n = 1..12, flattened</a>
%F A216648 T(n,k) = A216649(n-1,k)*2 + 2^(2*n-1) for n>1.
%e A216648 856 is element of row 5, the binary string representation (with totally balanced substrings enclosed in parentheses) is (1(10)(10)(1(10)0)0).  The encoded rooted tree is:
%e A216648 .    o
%e A216648 .   /|\
%e A216648 .  o o o
%e A216648 .      |
%e A216648 .      o
%e A216648 Triangle T(n,k) begins:
%e A216648 2;
%e A216648 12;
%e A216648 52,     56;
%e A216648 212,   216,  232,  240;
%e A216648 852,   856,  872,  880,  920,  936,  944,  976,  992;
%e A216648 3412, 3416, 3432, 3440, 3480, 3496, 3504, 3536, 3552, 3688, 3696, ...
%e A216648 Triangle T(n,k) in binary:
%e A216648 10;
%e A216648 1100;
%e A216648 110100,       111000;
%e A216648 11010100,     11011000,     11101000,     11110000;
%e A216648 1101010100,   1101011000,   1101101000,   1101110000,   1110011000, ...
%e A216648 110101010100, 110101011000, 110101101000, 110101110000, 110110011000, ...
%p A216648 F:= proc(n) option remember; `if`(n=1, [10], sort(map(h->
%p A216648       parse(cat(1, sort(h)[], 0)), g(n-1, n-1)))) end:
%p A216648 g:= proc(n, i) option remember; `if`(i=1, [[10$n]], [seq(seq(seq(
%p A216648       [seq (F(i)[w[t]-t+1], t=1..j),v[]], w=combinat[choose](
%p A216648       [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
%p A216648     end:
%p A216648 b:= proc(n) local h, i, r; h, r:= n, 0; for i from 0
%p A216648       while h>0 do r:= r+2^i*irem(h, 10, 'h') od; r
%p A216648     end:
%p A216648 T:= proc(n) option remember; map(b, F(n))[] end:
%p A216648 seq(T(n), n=1..7);
%Y A216648 First column gives: A080675.
%Y A216648 Last elements of rows give: A020522.
%Y A216648 Row lengths are: A000081.
%Y A216648 Subsequence of A057547, A081292.
%Y A216648 Cf. A061773, A216349, A216350, A216649.
%K A216648 nonn,tabf
%O A216648 1,1
%A A216648 _Alois P. Heinz_, Sep 12 2012

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