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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). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

FORMULA

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

EXAMPLE

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, ...

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);

CROSSREFS

First column gives: A080675.

Last elements of rows give: A020522.

Row lengths are: A000081.

Subsequence of A057547, A081292.

Cf. A061773, A216349, A216350, A216649.

Sequence in context: A054667 A009537 A057547 * A043007 A300572 A176580

Adjacent sequences:  A216645 A216646 A216647 * A216649 A216650 A216651

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 12 2012

STATUS

approved

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Last modified March 20 09:35 EDT 2019. Contains 321345 sequences. (Running on oeis4.)