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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. 10
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A219272 A084097 A293991 * A261494 A168377 A122867

Adjacent sequences:  A288635 A288636 A288637 * A288639 A288640 A288641

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 12 2017

STATUS

approved

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Last modified January 20 16:38 EST 2019. Contains 319335 sequences. (Running on oeis4.)