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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

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(y*(32-y))/(2*Pi*(81-y)),y=0..32) on (0,32): a(n+1) = int(x^n*W(x), x=0..32), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)

Example

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}).

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);

Mathematica

CoefficientList[Series[8/9 + 16/(9 (7 + 9 Sqrt[1 - 32 x])), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 16 2015 *)

Crossrefs

Column k=9 of A183134.

Cf. A194723, A194726, A174728.

Sequence in context: A171860 A324449 A191589 * A081421 A121824 A120287

Adjacent sequences: A194726 A194727 A194728 * A194730 A194731 A194732

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved