0,8

Alois P. Heinz, Table of n, a(n) for n = 0..350

a(n) = (A369695(n) - A000041(n))/2.

a(5) = 1: (221, 2111).

a(6) = 1: (2211, 21111).

a(7) = 4: (22111, 211111), (2221, 211111), (2221, 22111), (331, 31111).

a(8) = 5: (221111, 2111111), (22211, 2111111), (22211, 221111), (3221, 32111), (3311, 311111).

a(9) = 12: (2211111, 21111111), (222111, 21111111), (222111, 2211111), (22221, 21111111), (22221, 2211111), (22221, 222111), (32211, 321111), (33111, 3111111), (3321, 321111), (3321, 32211), (4221, 42111), (441, 411111).

a(10) = 19: (22111111, 211111111), (2221111, 211111111), (2221111, 22111111), (222211, 211111111), (222211, 22111111), (222211, 2221111), (322111, 3211111), (32221, 3211111), (32221, 322111), (331111, 31111111), (33211, 3211111), (33211, 322111), (33211, 32221), (3331, 31111111), (3331, 331111), (42211, 421111), (4411, 4111111), (442, 4222), (5221, 52111).

b:= proc(n, m, i) option remember; `if`(n=0,

`if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(

b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))

end:

a:= n-> (b(n$3)-combinat[numbpart](n))/2:

seq(a(n), n=0..50);

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1] + Sum[Sum[b[Sequence @@ Sort[{n-i*j, m-i*h}], i-1], {h, 1, m/i}], { j, 1, n/i}]]];

a[n_] := (b[n, n, n] - PartitionsP[n])/2;

Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

Cf. A000041, A297388, A369695, A369696.

Sequence in context: A054451 A309479 A352720 * A308775 A305335 A123102

Adjacent sequences: A369694 A369695 A369696 * A369698 A369699 A369700

Alois P. Heinz, Jan 29 2024

approved