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

0,3

Links

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

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)

Example

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

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);

Crossrefs

Column k=6 of A183134.

Sequence in context: A293610 A061613 A093750 * A367790 A296143 A217722

Adjacent sequences: A194723 A194724 A194725 * A194727 A194728 A194729

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved