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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.

%I A293580 #15 Dec 29 2020 10:48:41
%S A293580 13,132,924,5546,30720,162396,834004,4204080,20932656,103365416,
%T A293580 507538320,2482394448,12108785680,58954149792,286654114176,
%U A293580 1392524616032,6760326357888,32804684941248,159135076864576,771789378620928,3742512930335232,18145949724380288
%N 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.
%H A293580 Alois P. Heinz, <a href="/A293580/b293580.txt">Table of n, a(n) for n = 3..1000</a>
%F A293580 From _Vaclav Kotesovec_, Oct 14 2017: (Start)
%F A293580 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).
%F A293580 a(n) ~ (1 + 2^(1/3) + 2^(2/3))/6 * (2 + 2^(1/3) + 2^(2/3))^n. (End)
%p A293580 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A293580       add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
%p A293580     end:
%p A293580 a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
%p A293580 seq(a(n), n=3..30);
%t A293580 b[n_, k_] := b[n, k] = If[n == 0, 1,
%t A293580     Sum[b[n - j, k] Binomial[j + k - 1, k - 1], {j, 1, n}]];
%t A293580 a[n_] := With[{k = 3}, Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A293580 a /@ Range[3, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A293580 Column k=3 of A261781.
%K A293580 nonn
%O A293580 3,1
%A A293580 _Alois P. Heinz_, Oct 12 2017