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A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4).

%I A304941 #18 Jun 07 2023 19:44:03
%S A304941 1,6,18,68,246,948,3572,13896,53286,208452,807132,3169080,12346300,
%T A304941 48602760,190150440,750018448,2943363078,11627329764,45736940364,
%U A304941 180897649368,712881236052,2822389182104,11138924119512,44137230865392,174405194802524,691557285091176
%N A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4).
%C A304941 Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ... then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.
%H A304941 Seiichi Manyama, <a href="/A304941/b304941.txt">Table of n, a(n) for n = 0..1000</a>
%F A304941 n*a(n) = 6*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
%F A304941 a(n) ~ 2^(2*n + 3/4) / (Gamma(3/4) * n^(1/4)). - _Vaclav Kotesovec_, May 28 2018
%t A304941 CoefficientList[Series[((1+4x)/(1-4x))^(3/4),{x,0,30}],x] (* _Harvey P. Dale_, Oct 24 2020 *)
%o A304941 (PARI) N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(3/4))
%o A304941 (Magma) [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 8*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // _G. C. Greubel_, Jun 07 2023
%o A304941 (SageMath)
%o A304941 @CachedFunction
%o A304941 def a(n): # a = A304941
%o A304941     if n<2: return 6^n
%o A304941     else: return 2*(3*a(n-1) + 8*(n-2)*a(n-2))//n
%o A304941 [a(n) for n in range(41)] # _G. C. Greubel_, Jun 07 2023
%Y A304941 ((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), A304940 (m=2), this sequence (m=3), A081654 (m=4).
%K A304941 nonn
%O A304941 0,2
%A A304941 _Seiichi Manyama_, May 22 2018