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Number of quaternary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.
4

%I #23 Jul 19 2017 20:53:35

%S 1,1,7,58,523,4966,48838,492724,5068915,52955950,560198962,5987822380,

%T 64563867454,701383563388,7668869344108,84326618668648,

%U 931894610845123,10344218506421758,115280448164645818,1289346114476360188,14467472108268263818,162816535672067515828

%N Number of quaternary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

%H Alois P. Heinz, <a href="/A194724/b194724.txt">Table of n, a(n) for n = 0..500</a>

%H C. Kassel and C. Reutenauer, <a href="https://arxiv.org/abs/1303.3481">Algebraicity of the zeta function associated to a matrix over a free group algebra</a>, arXiv preprint arXiv:1303.3481, 2013

%F G.f.: 3/4 + 3/(2*(2+4*sqrt(1-12*x))).

%F a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*3^j for n>0.

%F a(n) ~ 3 * 12^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014

%F Conjecture: n*a(n) +2*(-14*n+9)*a(n-1) +96*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Mar 14 2015

%e a(2) = 7: aaaa, aabb, aacc, aadd, abba, acca, adda (with quaternary alphabet {a,b,c,d}).

%p a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *3^j, j=0..n-1)/n):

%p seq(a(n), n=0..25);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<3, [1, 1, 7][n+1],

%p ((28*n-18)*a(n-1) -(192*n-288)*a(n-2))/n)

%p end:

%p seq(a(n), n=0..30);

%t CoefficientList[Series[3/4+3/(2(2+4Sqrt[1-12x])),{x,0,30}],x] (* _Harvey P. Dale_, Sep 30 2012 *)

%Y Column k=4 of A183134.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 02 2011