A293580 Number of compositions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order and all three letters occur at least once in the composition.

13, 132, 924, 5546, 30720, 162396, 834004, 4204080, 20932656, 103365416, 507538320, 2482394448, 12108785680, 58954149792, 286654114176, 1392524616032, 6760326357888, 32804684941248, 159135076864576, 771789378620928, 3742512930335232, 18145949724380288

Offset: 3

Author

Alois P. Heinz, Oct 12 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A261781.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
seq(a(n), n=3..30);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1,
    Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];
a[n_] := With[{k = 3}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

From Vaclav Kotesovec, Oct 14 2017: (Start)

a(n) = 12*a(n-1) - 52*a(n-2) + 102*a(n-3) - 96*a(n-4) + 44*a(n-5) - 8*a(n-6).

a(n) ~ (1 + 2^(1/3) + 2^(2/3))/6 * (2 + 2^(1/3) + 2^(2/3))^n. (End)