A288693 - OEIS

%I #12 May 17 2022 04:46:38

%S 1,1,3,19,201,2841,48245,939863,20341201,487875964,12830282835,

%T 370205055144,11629998323185,396693714869323,14593231979817751,

%U 576427808563042857

%N Number of n-digit biquanimous strings using digits {0,1,...,n}.

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

%e a(2) = 3: 00, 11, 22.

%e a(3) = 19: 000, 011, 022, 033, 101, 110, 112, 121, 123, 132, 202, 211, 213, 220, 231, 303, 312, 321, 330.

%p b:= proc(n, k, s) option remember;

%p       `if`(n=0, `if`(s={}, 0, 1), add(b(n-1, k, select(y->

%p        y<=(n-1)*k, map(x-> [abs(x-i), x+i][], s))), i=0..k))

%p     end:

%p a:= n-> b(n$2, {0}):

%p seq(a(n), n=0..10);

%t 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 a[n_] := b[n, n, {0}];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 10}] (* _Jean-François Alcover_, May 17 2022, after _Alois P. Heinz_ *)

%Y Main diagonal of A288638.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jun 13 2017