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A321031 Number of words of length 3n such that the index set of occurring letters is {1, 2, ..., k}, all letters are introduced in ascending order, and the words can be built by repeatedly inserting triples into the initially empty word.

%I A321031 #9 May 18 2020 07:52:05
%S A321031 1,1,4,31,351,5144,91816,1918578,45687682,1216354021,35689352250,
%T A321031 1141323078031,39429988969021,1461049507764175,57720478019188989,
%U A321031 2419008380691088543,107083662651332423339,4988596265684542112304,243781041304397011647766
%N A321031 Number of words of length 3n such that the index set of occurring letters is {1, 2, ..., k}, all letters are introduced in ascending order, and the words can be built by repeatedly inserting triples into the initially empty word.
%H A321031 Alois P. Heinz, <a href="/A321031/b321031.txt">Table of n, a(n) for n = 0..411</a>
%e A321031 a(2) = 4: aaaaaa, aaabbb, aabbba, abbbaa.
%p A321031 a:= n-> `if`(n=0, 1, add(add((-1)^i*(k-i)/n*add(binomial(3*n, j)
%p A321031       *(n-j)*(k-i-1)^j, j=0..n-1)/(i!*(k-i)!), i=0..k), k=0..n)):
%p A321031 seq(a(n), n=0..20);
%Y A321031 Row sums of A256311.
%Y A321031 Cf. A258498.
%K A321031 nonn
%O A321031 0,3
%A A321031 _Alois P. Heinz_, Oct 26 2018