# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a194727 Showing 1-1 of 1 %I A194727 #26 Feb 08 2021 06:23:33 %S A194727 1,1,13,205,3565,65821,1265677,25066621,507709165,10466643805, %T A194727 218878998733,4631531585341,98980721277613,2133274258946845, %U A194727 46313701181477005,1011889827742935805,22232378278653590125,490899296804667191005,10887346288742800406605 %N A194727 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 A194727 Alois P. Heinz, Table of n, a(n) for n = 0..500 %F A194727 G.f.: 6/7 + 12/(7*(5+7*sqrt(1-24*x))). %F A194727 a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*6^j for n>0. %F A194727 a(n) ~ 6 * 24^n / (25 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014 %F A194727 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 A194727 From _Karol A. Penson_, Jul 16 2015: (Start) %F A194727 Special values of the hypergeometric function 2F1, in Maple notation: %F A194727 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 A194727 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 A194727 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 A194727 a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *6^j, j=0..n-1)/n): %p A194727 seq(a(n), n=0..20); %p A194727 # second Maple program: %p A194727 a:= proc(n) option remember; `if`(n<3, [1, 1, 13][n+1], %p A194727 ((73*n-36)*a(n-1) -(1176*n-1764)*a(n-2))/n) %p A194727 end: %p A194727 seq(a(n), n=0..30); %Y A194727 Column k=7 of A183134. %Y A194727 Cf. A194728, A194729, A194723. %K A194727 nonn %O A194727 0,3 %A A194727 _Alois P. Heinz_, Sep 02 2011 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE