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

Author

Alois P. Heinz, Jun 12 2017

Keywords

• nonn
• tabl

Comments

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

Examples

			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, ...
		

Links

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

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.

Programs

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 *)