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%I #26 Feb 08 2021 06:23:33
%S 1,1,13,205,3565,65821,1265677,25066621,507709165,10466643805,
%T 218878998733,4631531585341,98980721277613,2133274258946845,
%U 46313701181477005,1011889827742935805,22232378278653590125,490899296804667191005,10887346288742800406605
%N Number of 7-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 Alois P. Heinz, <a href="/A194727/b194727.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: 6/7 + 12/(7*(5+7*sqrt(1-24*x))).
%F a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*6^j for n>0.
%F a(n) ~ 6 * 24^n / (25 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014
%F D-finite with recurrence n*a(n) +(-73*n+36)*a(n-1) +588*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Mar 14 2015
%F From _Karol A. Penson_, Jul 16 2015: (Start)
%F Special values of the hypergeometric function 2F1, in Maple notation:
%F a(n+1) = (12/7)^2*24^n*GAMMA(n+3/2)*hypergeom([1,n+3/2],[n+3],24/49)/(sqrt(Pi)*(n+2)!), n=0,1,... .
%F Integral representation as the n-th moment of a positive function W(x) = sqrt(x*(24-x))/(2*Pi*(49-x)) on (0,24): a(n+1) = int(x^n*W(x), x=0..24), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)
%e a(2) = 13: aaaa, aabb, aacc, aadd, aaee, aaff, aagg, abba, acca, adda, aeea, affa, agga (with 7-ary alphabet {a,b,c,d,e,f,g}).
%p a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *6^j, j=0..n-1)/n):
%p seq(a(n), n=0..20);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<3, [1, 1, 13][n+1],
%p ((73*n-36)*a(n-1) -(1176*n-1764)*a(n-2))/n)
%p end:
%p seq(a(n), n=0..30);
%Y Column k=7 of A183134.
%Y Cf. A194728, A194729, A194723.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 02 2011
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