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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.

%I A288638 #20 Oct 18 2018 15:56:00
%S A288638 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,
%T A288638 92,106,32,1,1,1,7,46,201,421,333,64,1,1,1,8,64,376,1206,1830,1030,
%U A288638 128,1,1,1,9,85,633,2841,6751,7687,3153,256,1
%N 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.
%C A288638 A biquanimous string is a string whose digits can be split into two groups with equal sums.
%H A288638 Alois P. Heinz, <a href="/A288638/b288638.txt">Antidiagonals n = 0..30, flattened</a>
%e A288638 A(2,2) = 3: 00, 11, 22.
%e A288638 A(3,2) = 10: 000, 011, 022, 101, 110, 112, 121, 202, 211, 220.
%e A288638 A(3,3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.
%e A288638 A(4,1) = 8: 0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111.
%e A288638 Square array A(n,k) begins:
%e A288638   1,  1,    1,    1,     1,      1,      1,      1, ...
%e A288638   1,  1,    1,    1,     1,      1,      1,      1, ...
%e A288638   1,  2,    3,    4,     5,      6,      7,      8, ...
%e A288638   1,  4,   10,   19,    31,     46,     64,     85, ...
%e A288638   1,  8,   33,   92,   201,    376,    633,    988, ...
%e A288638   1, 16,  106,  421,  1206,   2841,   5801,  10696, ...
%e A288638   1, 32,  333, 1830,  6751,  19718,  48245, 104676, ...
%e A288638   1, 64, 1030, 7687, 36051, 128535, 372345, 939863, ...
%p A288638 b:= proc(n, k, s) option remember;
%p A288638       `if`(n=0, `if`(s={}, 0, 1), add(b(n-1, k, select(y->
%p A288638        y<=(n-1)*k, map(x-> [abs(x-i), x+i][], s))), i=0..k))
%p A288638     end:
%p A288638 A:= (n, k)-> b(n, k, {0}):
%p A288638 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A288638 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}]];
%t A288638 A[n_, k_] := b[n, k, {0}];
%t A288638 Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jun 08 2018, from Maple *)
%Y A288638 Columns k=0-9 give: A000012, A011782, A053156, A288687, A288688, A288689, A288690, A288691, A288692, A065024.
%Y A288638 Rows n=0+1,2-3 give: A000012, A000027(k+1), A005448(k+1).
%Y A288638 Main diagonal gives A288693.
%Y A288638 Cf. A064544, A064914.
%K A288638 nonn,tabl
%O A288638 0,9
%A A288638 _Alois P. Heinz_, Jun 12 2017