A216649 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 such that all consecutive totally balanced substrings are in nondecreasing order; n>=1, 1<=k<=A000081(n+1).

2, 10, 12, 42, 44, 52, 56, 170, 172, 180, 184, 204, 212, 216, 232, 240, 682, 684, 692, 696, 716, 724, 728, 744, 752, 820, 824, 852, 856, 872, 880, 920, 936, 944, 976, 992, 2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, 2900, 2904, 2920, 2928

Offset

1,1

Comments

There is a simple bijection between the elements of row n and the rooted trees with n+1 nodes. The tree has a root node. Each matching pair (1,0) in the binary string representation encodes an additional node, the totally balanced substrings encode lists of subtrees.

Links

Alois P. Heinz, Rows n = 1..11, flattened

Formula

T(n,k) = A216648(n+1,k)/2 - 2^(2*n).

Example

172 is element of row 4, the binary string representation (with totally balanced substrings enclosed in parentheses) is (10)(10)(1(10)0).  The encoded rooted tree is:
.    o
.   /|\
.  o o o
.      |
.      o
Triangle T(n,k) begins:
2;
10,     12;
42,     44,   52,   56;
170,   172,  180,  184,  204,  212,  216,  232,  240;
682,   684,  692,  696,  716,  724,  728,  744,  752,  820,  824, ...
2730, 2732, 2740, 2744, 2764, 2772, 2776, 2792, 2800, 2868, 2872, ...
Triangle T(n,k) in binary:
10;
1010,       1100;
101010,     101100,     110100,     111000;
10101010,   10101100,   10110100,   10111000,   11001100,   11010100, ...
1010101010, 1010101100, 1010110100, 1010111000, 1011001100, 1011010100, ...

Maple

F:= proc(n) option remember; `if`(n=1, [10], sort(map(h->
      parse(cat(1, sort(h)[], 0)), g(n-1, n-1)))) end:
g:= proc(n, i) option remember; `if`(i=1, [[10$n]], [seq(seq(seq(
      [seq (F(i)[w[t]-t+1], t=1..j), v[]], w=combinat[choose](
      [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
    end:
b:= proc(n) local h, i, r; h, r:= n/10, 0; for i from 0
      while h>1 do r:= r+2^i*irem(h, 10, 'h') od; r
    end:
T:= proc(n) option remember; map(b, F(n+1))[] end:
seq(T(n), n=1..6);

Crossrefs

First column gives: A020988.

Last elements of rows give: A020522.

Row lengths are: A000081(n+1).

Subsequence of A014486, A031443.

Cf. A061773, A216349, A216350, A216648.

Sequence in context: A035928 A014486 A166751 * A071162 A075165 A209641

Adjacent sequences: A216646 A216647 A216648 * A216650 A216651 A216652

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 12 2012

Status

approved