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).

2, 12, 52, 56, 212, 216, 232, 240, 852, 856, 872, 880, 920, 936, 944, 976, 992, 3412, 3416, 3432, 3440, 3480, 3496, 3504, 3536, 3552, 3688, 3696, 3752, 3760, 3792, 3808, 3888, 3920, 3936, 4000, 4032, 13652, 13656, 13672, 13680, 13720, 13736, 13744, 13776

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 nodes. Each matching pair (1,0) in the binary string representation encodes a node, each totally balanced substring encodes a list of subtrees.

Examples

			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:
.    o
.   /|\
.  o o o
.      |
.      o
Triangle T(n,k) begins:
2;
12;
52,     56;
212,   216,  232,  240;
852,   856,  872,  880,  920,  936,  944,  976,  992;
3412, 3416, 3432, 3440, 3480, 3496, 3504, 3536, 3552, 3688, 3696, ...
Triangle T(n,k) in binary:
10;
1100;
110100,       111000;
11010100,     11011000,     11101000,     11110000;
1101010100,   1101011000,   1101101000,   1101110000,   1110011000, ...
110101010100, 110101011000, 110101101000, 110101110000, 110110011000, ...
		

Links

• Alois P. Heinz, Rows n = 1..12, flattened

Crossrefs

First column gives: A080675.

Last elements of rows give: A020522.

Row lengths are: A000081.

Subsequence of A057547, A081292.

Cf. A061773, A216349, A216350, A216649.

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, 0; for i from 0
      while h>0 do r:= r+2^i*irem(h, 10, 'h') od; r
    end:
T:= proc(n) option remember; map(b, F(n))[] end:
seq(T(n), n=1..7);

Formula

T(n,k) = A216649(n-1,k)*2 + 2^(2*n-1) for n>1.