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

3,1

Links

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

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)

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 *)

Crossrefs

Column k=3 of A261781.

Sequence in context: A037582 A065539 A048554 * A295349 A161498 A037715

Adjacent sequences: A293577 A293578 A293579 * A293581 A293582 A293583

Keyword

nonn

Author

Alois P. Heinz, Oct 12 2017

Status

approved