A194727 Number of 7-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, 13, 205, 3565, 65821, 1265677, 25066621, 507709165, 10466643805, 218878998733, 4631531585341, 98980721277613, 2133274258946845, 46313701181477005, 1011889827742935805, 22232378278653590125, 490899296804667191005, 10887346288742800406605

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

Column k=7 of A183134.

Cf. A194728, A194729, A194723.

Sequence in context: A057804 A215621 A299960 * A059355 A243783 A206348

Adjacent sequences: A194724 A194725 A194726 * A194728 A194729 A194730

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved