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%I #28 Aug 09 2023 13:13:24
%S 1,2,12,68,396,2358,14262,87252,538440,3345434,20899816,131154264,
%T 826135794,5220372274,33077821314,210087769632,1337104370320,
%U 8525602760550,54449281992528,348250972411252,2230296171922008,14300414859019290,91791793780179790
%N Expansion of 1/sqrt(1 - 4*x/(1-x)^3).
%H Winston de Greef, <a href="/A359489/b359489.txt">Table of n, a(n) for n = 0..1218</a>
%F a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+2*k-1,n-k).
%F n*a(n) = (8*n-6)*a(n-1) - (10*n-24)*a(n-2) + 4*(n-3)*a(n-3) - (n-4)*a(n-4) for n > 3.
%F a(n) ~ sqrt(2*(2 + (35 + 3*sqrt(129))^(1/3))) * (40 + 7*(262 + 6*sqrt(129))^(1/3) + (262 + 6*sqrt(129))^(2/3))^n / ((43*(86 + 6*sqrt(129)))^(1/6) * sqrt(Pi*n) * 3^n * (262 + 6*sqrt(129))^(n/3)). - _Vaclav Kotesovec_, Mar 25 2023
%F a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+1-k,2) * a(k). - _Seiichi Manyama_, Mar 28 2023
%t CoefficientList[Series[1/Sqrt[1-(4x)/(1-x)^3],{x,0,30}],x] (* _Harvey P. Dale_, Aug 09 2023 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x)^3))
%o (PARI) a(n) = sum(k=0, n, binomial(2*k,k) * binomial(n+2*k-1,n-k)) \\ _Winston de Greef_, Mar 24 2023
%Y Cf. A085362, A110170, A162478, A359758, A360132.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Mar 24 2023
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