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

Author

Alois P. Heinz, Sep 12 2012

Keywords

• nonn
• tabf

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.

Examples

			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, ...
		

Links

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

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.

Programs

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);

Formula

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