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A293367 Number of partitions 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 partition.

%I A293367 #12 Dec 08 2020 08:36:25
%S A293367 10,81,396,1751,6528,23892,80979,272085,876342,2821217,8840964,
%T A293367 27713589,85532512,263935014,806417553,2464692788,7483544643,
%U A293367 22727335830,68734242687,207887123472,627024671262,1891376241178,5694616254570,17146333061406,51564199968339
%N A293367 Number of partitions 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 partition.
%H A293367 Alois P. Heinz, <a href="/A293367/b293367.txt">Table of n, a(n) for n = 3..1000</a>
%F A293367 a(n) ~ c * 3^n, where c = 6.846206073498521357898163368676070142316815386135993166380819930419737... - _Vaclav Kotesovec_, Oct 11 2017
%p A293367 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293367       b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p A293367     end:
%p A293367 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
%p A293367 seq(a(n), n=3..30);
%t A293367 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]]]];
%t A293367 a[n_] := With[{k = 3}, Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A293367 a /@ Range[3, 30] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A293367 Column k=3 of A261719.
%Y A293367 Cf. A261737.
%K A293367 nonn
%O A293367 3,1
%A A293367 _Alois P. Heinz_, Oct 07 2017