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

2,1

Links

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

Formula

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

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 *)

Crossrefs

Column k=2 of A261719.

Sequence in context: A373301 A032093 A007993 * A327319 A080929 A052482

Adjacent sequences: A293363 A293364 A293365 * A293367 A293368 A293369

Keyword

nonn

Author

Alois P. Heinz, Oct 07 2017

Status

approved