A293366 Number of partitions of n where each part i is marked with a word of length i over a binary alphabet whose letters appear in alphabetical order and both letters occur at least once in the partition.

3, 12, 40, 104, 279, 654, 1577, 3560, 8109, 17734, 39205, 83996, 181043, 382856, 811084, 1694468, 3545864, 7340308, 15205768, 31259422, 64253260, 131314502, 268332975, 545854344, 1110087515, 2250051262, 4558868119, 9213241988, 18613362500, 37529700206

Offset: 2

Author

Alois P. Heinz, Oct 07 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=2 of A261719.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..35);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 35] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 2^n, where c = A256155 = 18.563146563610114727475354232269284... - Vaclav Kotesovec, Oct 11 2017