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%I #15 Jan 30 2020 21:29:15
%S 1,1,5,9,37,89,325,905,3109,9337,31173,97449,321445,1027225,3374405,
%T 10920649,35855909,116937145,384340421,1259728873,4147000229,
%U 13639616473,44978045765,148314302473,489879442469,1618600915705
%N Expansion of 1/sqrt(1-2*x-7*x^2+8*x^3).
%C 1/sqrt(1-2*x-(4*r-1)*x^2+4*r^3) expands to give Sum_{k=0..floor(n/2)} binomial(2*k,k)*binomial(n-k,n-2*k)*r^k.
%H G. C. Greubel, <a href="/A098477/b098477.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*k,k)*binomial(n-k, n-2k)*2^k.
%F G.f.: 1/(1-x-4x^2/(1-0x-2x^2/(1-x-2x^2/(1-0x-2x^2/(1-x-2x^2/.... (continued fraction). - _Paul Barry_, Dec 07 2008
%F D-finite with recurrence: n*a(n) +(1-2*n)*a(n-1) +7*(1-n)*a(n-2) +4*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 09 2012
%F a(n) ~ 16 * ((1+sqrt(33))/2)^n / (sqrt(594-50*sqrt(33)) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 04 2014
%t CoefficientList[Series[1/Sqrt[1-2*x-7*x^2+8*x^3], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 04 2014 *)
%o (PARI) x='x+O('x^50); Vec(1/sqrt(1-2*x-7*x^2+8*x^3)) \\ _G. C. Greubel_, Mar 16 2017
%Y Cf. A026569, A098478.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 10 2004
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