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A194728 Number of 8-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. 5
1, 1, 15, 274, 5531, 118686, 2654646, 61189668, 1443039123, 34648845862, 844131474530, 20813234394492, 518373091849502, 13021801045587244, 329543346098061516, 8393705745623980104, 215009056951891319811, 5535306699430995140214, 143144289829339089562986 (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.: 7/8 + 7/(4*(6+8*sqrt(1-28*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*7^j for n>0.

a(n) ~ 7 * 28^n / (36 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

n*a(n) +2*(-46*n+21)*a(n-1) +896*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 14 2015

From Karol A. Penson, Jul 13 2015: (Start)

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

a(n+1) = (7/4)^2*(28)^n*GAMMA(n+3/2)*hypergeom([1, n+3/2], [n+3],7/16)/(sqrt(Pi)*(n+2)!), n=0,1,... .

Integral representation as the n-th moment of a positive function W(x) = sqrt((28-x)*x)*(1/(64-x))/(2*Pi) on (0,28): a(n+1) = int(x^n*W(x), x=0..28), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)

EXAMPLE

a(2) = 15: aaaa, aabb, aacc, aadd, aaee, aaff, aagg, aahh, abba, acca, adda, aeea, affa, agga, ahha (with 8-ary alphabet {a,b,c,d,e,f,g,h}).

MAPLE

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *7^j, j=0..n-1) /n):

seq(a(n), n=0..20);

CROSSREFS

Column k=8 of A183134.

Cf. A194723, A194726.

Sequence in context: A227973 A195615 A156091 * A284077 A034687 A159239

Adjacent sequences:  A194725 A194726 A194727 * A194729 A194730 A194731

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 02 2011

STATUS

approved

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Last modified October 15 15:20 EDT 2018. Contains 316236 sequences. (Running on oeis4.)