OFFSET
0,2
COMMENTS
Sixth column of array A103209.
The Hankel transform of this sequence is 30^C(n+1,2). - Philippe Deléham, Oct 28 2007
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..745
FORMULA
G.f.: (1-z-sqrt(z^2-22*z+1))/(10*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*5^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*5^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2*k) 5^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 5*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 11*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 3^(1/4) * (11 + 2*sqrt(30))^(n + 1/2) / (10^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2021
MATHEMATICA
CoefficientList[Series[(1-x-Sqrt[x^2-22*x+1])/(10*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)
PROG
(PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-22*x+1))/(10*x)) \\ G. C. Greubel, Feb 10 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-22*x+1))/(10*x))) // G. C. Greubel, Feb 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Oct 18 2007
STATUS
approved