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A130976
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G.f.: 8/(3 + 5*sqrt(1-16*x)).
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6
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1, 5, 45, 485, 5725, 71445, 925965, 12335685, 167817405, 2321105525, 32536755565, 461181239205, 6598203881245, 95157851939285, 1381842797170125, 20187779510360325, 296499276685062525, 4375281190871356725, 64836419120040890925
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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, ...
...
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
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)
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MAPLE
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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);
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MATHEMATICA
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CoefficientList[Series[8/(3+5*Sqrt[1-16*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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