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A194724 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.

%I A194724 #23 Jul 19 2017 20:53:35
%S A194724 1,1,7,58,523,4966,48838,492724,5068915,52955950,560198962,5987822380,
%T A194724 64563867454,701383563388,7668869344108,84326618668648,
%U A194724 931894610845123,10344218506421758,115280448164645818,1289346114476360188,14467472108268263818,162816535672067515828
%N A194724 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 A194724 Alois P. Heinz, <a href="/A194724/b194724.txt">Table of n, a(n) for n = 0..500</a>
%H A194724 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 A194724 G.f.: 3/4 + 3/(2*(2+4*sqrt(1-12*x))).
%F A194724 a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*3^j for n>0.
%F A194724 a(n) ~ 3 * 12^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014
%F A194724 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 A194724 a(2) = 7: aaaa, aabb, aacc, aadd, abba, acca, adda (with quaternary alphabet {a,b,c,d}).
%p A194724 a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *3^j, j=0..n-1)/n):
%p A194724 seq(a(n), n=0..25);
%p A194724 # second Maple program:
%p A194724 a:= proc(n) option remember; `if`(n<3, [1, 1, 7][n+1],
%p A194724       ((28*n-18)*a(n-1) -(192*n-288)*a(n-2))/n)
%p A194724     end:
%p A194724 seq(a(n), n=0..30);
%t A194724 CoefficientList[Series[3/4+3/(2(2+4Sqrt[1-12x])),{x,0,30}],x] (* _Harvey P. Dale_, Sep 30 2012 *)
%Y A194724 Column k=4 of A183134.
%K A194724 nonn
%O A194724 0,3
%A A194724 _Alois P. Heinz_, Sep 02 2011