A194726 Number of 6-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, 11, 146, 2131, 32966, 530526, 8786436, 148733571, 2561439806, 44731364266, 790211926076, 14095578557486, 253519929631996, 4592415708939356, 83709533881191816, 1534227271236577251, 28256420350942562286, 522675506718404898546, 9706083027629177910156

Offset: 0

Author

Alois P. Heinz, Sep 02 2011

Keywords

• nonn

Examples

			a(2) = 11: aaaa, aabb, aacc, aadd, aaee, aaff, abba, acca, adda, aeea, affa (with 6-ary alphabet {a,b,c,d,e,f}).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Column k=6 of A183134.

Programs

Maple

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *5^j, j=0..n-1) /n):
seq(a(n), n=0..25);

Formula

G.f.: 5/6 + 5/(3*(4+6*sqrt(1-20*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*5^j for n>0.

a(n) ~ 5*20^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

D-finite with recurrence: n*a(n) = 2*(28*n-15)*a(n-1) - 360*(2*n-3)*a(n-2). - Vaclav Kotesovec, Aug 13 2013

From Karol A. Penson, Jul 12 2015: (Start)

Special values of the hypergeometric function 2F1, in Maple notation:

a(n+1) = (25/9)*20^n*GAMMA(n+3/2)*hypergeom([1, n+3/2], [n+3],5/9)/(sqrt(Pi)*(n+2)!), n=0,1,... .

Integral representation as the n-th moment of a positive function W(x) = sqrt((20-x)*x)*(1/(36-x))/(2*Pi) on (0,20): a(n+1) = int(x^n*W(x),x=0..20), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)