OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
G.f.: 6/7 + 12/(7*(5+7*sqrt(1-24*x))).
a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*6^j for n>0.
a(n) ~ 6 * 24^n / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014
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
From Karol A. Penson, Jul 16 2015: (Start)
Special values of the hypergeometric function 2F1, in Maple notation:
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,... .
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)
EXAMPLE
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}).
MAPLE
a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *6^j, j=0..n-1)/n):
seq(a(n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 13][n+1],
((73*n-36)*a(n-1) -(1176*n-1764)*a(n-2))/n)
end:
seq(a(n), n=0..30);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 02 2011
STATUS
approved