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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.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 12 2017
STATUS
approved