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A194729 Number of 9-ary 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 A194729 #25 Feb 15 2025 20:22:01
%S A194729 1,1,17,353,8113,198401,5060433,133071009,3581326065,98156060225,
%T A194729 2730108129937,76862217117665,2186096427128369,62718004238927233,
%U A194729 1812849590253944273,52742324721313632033,1543272031837984426353,45386639860532255882433,1340844916965007902013713
%N A194729 Number of 9-ary 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 A194729 Alois P. Heinz, <a href="/A194729/b194729.txt">Table of n, a(n) for n = 0..200</a>
%F A194729 G.f.: 8/9 + 16/(9*(7+9*sqrt(1-32*x))).
%F A194729 a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*8^j for n>0.
%F A194729 a(n) ~ 8 * 32^n / (49 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014
%F A194729 n*a(n) +(-113*n+48)*a(n-1) +1296*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Mar 14 2015
%F A194729 From _Karol A. Penson_, Jul 15 2015: (Start)
%F A194729 Special values of the hypergeometric function 2F1, in Maple notation:
%F A194729 a(n+1) = 2^8*32^n*GAMMA(n+3/2)*hypergeom([1,n+3/2],[n+3],32/81)/(81*sqrt(Pi)*(n+2)!), n=0,1,... .
%F A194729 Integral representation as the n-th moment of a positive function W(x) = sqrt(x*(32-x))/(2*Pi*(81-x)) on (0,32): a(n+1) = Integral_{x=0..32} x^n*W(x) dx, n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)
%e A194729 a(2) = 17: aaaa, aabb, aacc, aadd, aaee, aaff, aagg, aahh, aaii, abba, acca, adda, aeea, affa, agga, ahha, aiia (with 9-ary alphabet {a,b,c,d,e,f,g,h,i}).
%p A194729 a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *8^j, j=0..n-1) /n):
%p A194729 seq(a(n), n=0..20);
%t A194729 CoefficientList[Series[8/9 + 16/(9 (7 + 9 Sqrt[1 - 32 x])), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jul 16 2015 *)
%Y A194729 Column k=9 of A183134.
%Y A194729 Cf. A194723, A194726, A174728.
%K A194729 nonn
%O A194729 0,3
%A A194729 _Alois P. Heinz_, Sep 02 2011