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%I #10 Oct 23 2015 12:21:09
%S 1,1,13,217,4085,82593,1751197,38413481,864413317,19842830065,
%T 462825376685,10937407206265,261311076852245,6301225556698177,
%U 153160687795008445,3748598210810053449,92303640047399410341,2285025852515378528913,56836898766186234593485
%N Number of 3n-length 5-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 Alois P. Heinz, <a href="/A218475/b218475.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*4^j for n>0, a(0) = 1.
%F Recurrence: n*(2*n-1)*(4*n-5)*a(n) = (1216*n^3 - 2452*n^2 + 1267*n - 120)*a(n-1) - 750*(3*n-5)*(3*n-4)*(4*n-1)*a(n-2). - _Vaclav Kotesovec_, Aug 31 2014
%F a(n) ~ 4 * 3^(3*n+1/2) / (49 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 31 2014
%p a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*4^j, j=0..n-1)/n):
%p seq(a(n), n=0..20);
%Y Column k=5 of A213027.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 29 2012
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