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.

1, 1, 15, 274, 5531, 118686, 2654646, 61189668, 1443039123, 34648845862, 844131474530, 20813234394492, 518373091849502, 13021801045587244, 329543346098061516, 8393705745623980104, 215009056951891319811, 5535306699430995140214, 143144289829339089562986

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

D-finite with recurrence 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 A339118 A034687

Adjacent sequences: A194725 A194726 A194727 * A194729 A194730 A194731

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved