A288638 Number A(n,k) of n-digit biquanimous strings using digits {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 33, 16, 1, 1, 1, 6, 31, 92, 106, 32, 1, 1, 1, 7, 46, 201, 421, 333, 64, 1, 1, 1, 8, 64, 376, 1206, 1830, 1030, 128, 1, 1, 1, 9, 85, 633, 2841, 6751, 7687, 3153, 256, 1

Offset

0,9

Comments

A biquanimous string is a string whose digits can be split into two groups with equal sums.

Links

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Example

A(2,2) = 3: 00, 11, 22.
A(3,2) = 10: 000, 011, 022, 101, 110, 112, 121, 202, 211, 220.
A(3,3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.
A(4,1) = 8: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111.
Square array A(n,k) begins:
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  1,    1,    1,     1,      1,      1,      1, ...
  1,  2,    3,    4,     5,      6,      7,      8, ...
  1,  4,   10,   19,    31,     46,     64,     85, ...
  1,  8,   33,   92,   201,    376,    633,    988, ...
  1, 16,  106,  421,  1206,   2841,   5801,  10696, ...
  1, 32,  333, 1830,  6751,  19718,  48245, 104676, ...
  1, 64, 1030, 7687, 36051, 128535, 372345, 939863, ...

Maple

b:= proc(n, k, s) option remember;
      `if`(n=0, `if`(s={}, 0, 1), add(b(n-1, k, select(y->
       y<=(n-1)*k, map(x-> [abs(x-i), x+i][], s))), i=0..k))
    end:
A:= (n, k)-> b(n, k, {0}):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[n_, k_, s_] := b[n, k, s] = If[n == 0, If[s == {}, 0, 1], Sum[b[n-1, k, Select[Flatten[{Abs[#-i], #+i}& /@ s], # <= (n-1)*k&]], {i, 0, k}]];
A[n_, k_] := b[n, k, {0}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 08 2018, from Maple *)

Crossrefs

Columns k=0-9 give: A000012, A011782, A053156, A288687, A288688, A288689, A288690, A288691, A288692, A065024.

Rows n=0+1,2-3 give: A000012, A000027(k+1), A005448(k+1).

Main diagonal gives A288693.

Cf. A064544, A064914.

Sequence in context: A084097 A306684 A293991 * A261494 A365673 A349574

Adjacent sequences: A288635 A288636 A288637 * A288639 A288640 A288641

Keyword

nonn,tabl

Author

Alois P. Heinz, Jun 12 2017

Status

approved