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.
1, 1, 17, 353, 8113, 198401, 5060433, 133071009, 3581326065, 98156060225, 2730108129937, 76862217117665, 2186096427128369, 62718004238927233, 1812849590253944273, 52742324721313632033, 1543272031837984426353, 45386639860532255882433, 1340844916965007902013713
Offset: 0
Keywords
Examples
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}).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Programs
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Maple
a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *8^j, j=0..n-1) /n): seq(a(n), n=0..20);
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Mathematica
CoefficientList[Series[8/9 + 16/(9 (7 + 9 Sqrt[1 - 32 x])), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 16 2015 *)
Formula
G.f.: 8/9 + 16/(9*(7+9*sqrt(1-32*x))).
a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*8^j for n>0.
a(n) ~ 8 * 32^n / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014
n*a(n) +(-113*n+48)*a(n-1) +1296*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 14 2015
From Karol A. Penson, Jul 15 2015: (Start)
Special values of the hypergeometric function 2F1, in Maple notation:
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,... .
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)