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A218474 Number of 3n-length 4-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.

%I A218474 #14 Dec 18 2020 11:51:16
%S A218474 1,1,10,127,1810,27631,441604,7293700,123485914,2131511455,
%T A218474 37368531010,663539143015,11908626395320,215670579863428,
%U A218474 3936425910379840,72335601620713432,1337149262553687658,24847762997547701695,463900901255209923310,8697278488612398979645
%N A218474 Number of 3n-length 4-ary words, either empty or beginning with the first letter of the alphabet, that can be built by repeatedly inserting triples of identical letters into the initially empty word.
%H A218474 Alois P. Heinz, <a href="/A218474/b218474.txt">Table of n, a(n) for n = 0..300</a>
%F A218474 a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*3^j for n>0, a(0) = 1.
%F A218474 a(n) ~ 3^(4*n+3/2) / (25*sqrt(Pi)*n^(3/2)*4^n). - _Vaclav Kotesovec_, Jul 16 2014
%p A218474 a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*3^j, j=0..n-1)/n):
%p A218474 seq(a(n), n=0..20);
%p A218474 # second Maple program:
%p A218474 a:= proc(n) option remember; `if`(n<3, [1, 1, 10][n+1],
%p A218474       ((2359*n^3 -5063*n^2 +2898*n -360)*a(n-1)
%p A218474        -576*(3*n-5)*(7*n-2)*(3*n-4)*a(n-2))/
%p A218474        (2*(2*n-1)*(7*n-9)*n))
%p A218474     end:
%p A218474 seq(a(n), n=0..30);
%t A218474 a[n_] := If[n == 0, 1, Sum[Binomial[3n, j] (n - j) 3^j, {j, 0, n - 1}]/n];
%t A218474 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 18 2020, after Maple *)
%Y A218474 Column k=4 of A213027.
%K A218474 nonn
%O A218474 0,3
%A A218474 _Alois P. Heinz_, Oct 29 2012