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Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).
4

%I #17 Apr 20 2018 09:03:34

%S 1,-3,-9,-288,459,-19278,-1539,-1265301,10734525,-147277926,520204923,

%T -7511358663,88687160577,-668191863951,5357547144702,-87542760890124,

%U 967961569696722,-5115624735401361,46065749188891275,-430898393089547667,6203508335817169257

%N Expansion of Product_{n>=1} (1 + (9*x)^n)^(-1/3).

%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = -9^n.

%H Seiichi Manyama, <a href="/A303130/b303130.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/18)) * 3^(2*n - 1/2) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 20 2018

%t CoefficientList[Series[(2/QPochhammer[-1, 9*x])^(1/3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 20 2018 *)

%o (PARI) N=99; x='x+O('x^N); Vec(prod(k=1, N, (1 + (9*x)^k)^(-1/3))) \\ _Altug Alkan_, Apr 20 2018

%Y Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(-1/b): A081362 (b=1), A298993 (b=2), this sequence (b=3), A303131 (b=4), A303132 (b=5).

%Y Cf. A303074.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 19 2018