OFFSET
0,2
COMMENTS
Number of walks of length 2n on the 5-regular tree beginning and ending at some fixed vertex. Hankel transform is A135292. - Philippe Deléham, Feb 25 2009
Also the number of length 2n words over an alphabet of size 5 that can be built by repeatedly inserting doublets into the initially empty word.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.
FORMULA
a(n) = Sum_{k=0..n} A039599(n,k) * 4^(n-k). - Philippe Deléham, Aug 25 2007
a(0) = 1; a(n) = (5/n) * Sum_{j=0..n-1} C(2*n,j) * (n-j) * 4^j for n > 0.
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
5, 5, 0, 0, 0, 0, ...
4, 4, 4, 0, 0, 0, ...
4, 4, 4, 4, 0, 0, ...
4, 4, 4, 4, 4, 0, ...
4, 4, 4, 4, 4, 4, ...
...
- Gary W. Adamson, Jul 13 2011
D-finite with recurrence: n*a(n) = (41*n-24)*a(n-1) - 200*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 20*16^n/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
From Karol A. Penson, Jul 02 2015: (Start)
Special values of the hypergeometric function 2F1, in Maple notation:
a(n) = 4*16^n*GAMMA(n+1/2)*hypergeom([1, n+1/2], [n+2], 16/25)/(5*sqrt(Pi)*(n+1)!), n=0,1,...
Moment representation as the 2n-th moment of the positive function
W(x) = 5*sqrt(16-x^2)/(Pi*(25-x^2)) on (0,4):
a(n) = int(x^(2*n)*W(x),x=0..4), n=0,1,... . (End)
MAPLE
a:= n-> `if`(n=0, 1, 5/n*add(binomial(2*n, j) *(n-j)*4^j, j=0..n-1)):
seq(a(n), n=0..20);
MATHEMATICA
CoefficientList[Series[8/(3+5*Sqrt[1-16*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Aug 23 2007
EXTENSIONS
More terms from Olivier Gérard, Sep 22 2007
Edited by Alois P. Heinz, Jan 17 2011
STATUS
approved