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%I #48 Dec 03 2021 19:28:47
%S 1,5,45,485,5725,71445,925965,12335685,167817405,2321105525,
%T 32536755565,461181239205,6598203881245,95157851939285,
%U 1381842797170125,20187779510360325,296499276685062525,4375281190871356725,64836419120040890925
%N G.f.: 8/(3 + 5*sqrt(1-16*x)).
%C 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
%C 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.
%H Vincenzo Librandi, <a href="/A130976/b130976.txt">Table of n, a(n) for n = 0..200</a>
%H Libor Caha and Daniel Nagaj, <a href="https://arxiv.org/abs/1805.07168">The pair-flip model: a very entangled translationally invariant spin chain</a>, arXiv:1805.07168 [quant-ph], 2018.
%H Pakawut Jiradilok and Supanat Kamtue, <a href="https://arxiv.org/abs/2107.09876">Transportation Distance between Probability Measures on the Infinite Regular Tree</a>, arXiv:2107.09876 [math.CO], 2021.
%F a(n) = Sum_{k=0..n} A039599(n,k) * 4^(n-k). - _Philippe Deléham_, Aug 25 2007
%F a(0) = 1; a(n) = (5/n) * Sum_{j=0..n-1} C(2*n,j) * (n-j) * 4^j for n > 0.
%F a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
%F 5, 5, 0, 0, 0, 0, ...
%F 4, 4, 4, 0, 0, 0, ...
%F 4, 4, 4, 4, 0, 0, ...
%F 4, 4, 4, 4, 4, 0, ...
%F 4, 4, 4, 4, 4, 4, ...
%F ...
%F - _Gary W. Adamson_, Jul 13 2011
%F 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
%F a(n) ~ 20*16^n/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012
%F From _Karol A. Penson_, Jul 02 2015: (Start)
%F Special values of the hypergeometric function 2F1, in Maple notation:
%F 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,...
%F Moment representation as the 2n-th moment of the positive function
%F W(x) = 5*sqrt(16-x^2)/(Pi*(25-x^2)) on (0,4):
%F a(n) = int(x^(2*n)*W(x),x=0..4), n=0,1,... . (End)
%p a:= n-> `if`(n=0, 1, 5/n*add(binomial(2*n, j) *(n-j)*4^j, j=0..n-1)):
%p seq(a(n), n=0..20);
%t CoefficientList[Series[8/(3+5*Sqrt[1-16*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%Y Column k=5 of A183135.
%Y Cf. A007318.
%K nonn
%O 0,2
%A _Philippe Deléham_, Aug 23 2007
%E More terms from _Olivier Gérard_, Sep 22 2007
%E Edited by _Alois P. Heinz_, Jan 17 2011
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