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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. 3
1, 1, 13, 205, 3565, 65821, 1265677, 25066621, 507709165, 10466643805, 218878998733, 4631531585341, 98980721277613, 2133274258946845, 46313701181477005, 1011889827742935805, 22232378278653590125, 490899296804667191005, 10887346288742800406605 (list; graph; refs; listen; history; text; internal format)
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

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: A057807 A057804 A215621 * A059355 A243783 A206348

Adjacent sequences:  A194724 A194725 A194726 * A194728 A194729 A194730

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 02 2011

STATUS

approved

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Last modified February 18 02:09 EST 2018. Contains 299297 sequences. (Running on oeis4.)