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Expansion of ((1 + 4*x)/(1 - 4*x))^(1/4).
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%I #25 Nov 14 2023 02:10:46

%S 1,2,2,12,22,124,276,1496,3686,19436,51068,263720,724860,3681880,

%T 10466920,52450992,153093254,758495564,2261603564,11096526344,

%U 33676743956,163842737928,504738342808,2437418983888,7605947276508,36487283224952,115140704639576

%N Expansion of ((1 + 4*x)/(1 - 4*x))^(1/4).

%C Let ((1 + k*x)/(1 - k*x))^(1/k) = a(0) + a(1)*x + a(2)*x^2 + ...

%C Then n*a(n) = 2*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

%H Seiichi Manyama, <a href="/A303537/b303537.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 2^(2*n + 1/4) / (Gamma(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 26 2018

%F n*a(n) = 2*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.

%F G.f.: A(x)=F(x*G(x^2)), where F(x) is the g.f. for A063886, and G(x) is the g.f. for A151403. - _Alexander Burstein_, Nov 13 2023

%t CoefficientList[Series[Surd[(1+4x)/(1-4x),4],{x,0,40}],x] (* _Harvey P. Dale_, Jul 25 2021 *)

%o (PARI) N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/4))

%Y Cf. A063886, A081654, A151403, A303538.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 25 2018