|
%I #12 Oct 23 2015 12:17:11
%S 1,1,28,1027,42850,1929043,91320022,4480436836,225785025802,
%T 11617042380355,607729841261560,32227411217273515,1728444323307664720,
%U 93593058046710649012,5109705135623767855960,280954986758729989837624,15544627425243191634814666
%N Number of 3n-length 10-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.
%C In general, column k of A213027 is (for k > 1) asymptotic to a(n) ~ 3^(3*n+1/2) * (k-1)^(n+1) / (sqrt(Pi) * (2*k-3)^2 * 4^n * n^(3/2)). - _Vaclav Kotesovec_, Aug 31 2014
%H Alois P. Heinz, <a href="/A218480/b218480.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 1/n * Sum_{j=0..n-1} C(3*n,j)*(n-j)*9^j for n>0, a(0) = 1.
%F Recurrence: 2*n*(2*n-1)*(13*n-15)*a(n) = (55159*n^3 - 95963*n^2 + 38478*n - 1080)*a(n-1) - 27000*(3*n-5)*(3*n-4)*(13*n-2)*a(n-2). - _Vaclav Kotesovec_, Aug 31 2014
%F a(n) ~ 3^(5*n+5/2) / (289 * sqrt(Pi) * 4^n * n^(3/2)). - _Vaclav Kotesovec_, Aug 31 2014
%p a:= n-> `if`(n=0, 1, add(binomial(3*n, j)*(n-j)*9^j, j=0..n-1)/n):
%p seq(a(n), n=0..20);
%Y Column k=10 of A213027.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 29 2012
|
