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Expansion of Product_{n>=1} ((1 + 8*x^n)/(1 - 8*x^n))^(1/8).
4

%I #19 Apr 23 2018 18:54:40

%S 1,2,4,50,98,1830,4576,83950,236500,4211766,12903260,222377926,

%T 723722602,12136867530,41382435824,678060771778,2400028798290,

%U 38546050682278,140724756748476,2220907298526934,8323586858891766,129340015891714962,495838256186203600

%N Expansion of Product_{n>=1} ((1 + 8*x^n)/(1 - 8*x^n))^(1/8).

%H Seiichi Manyama, <a href="/A303382/b303382.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ c * 8^n / n^(7/8), where c = (QPochhammer[-1, 1/8] / QPochhammer[1/8])^(1/8) / Gamma(1/8) = 0.15003359366795844474467456149... - _Vaclav Kotesovec_, Apr 23 2018

%p seq(coeff(series(mul(((1+8*x^k)/(1-8*x^k))^(1/8), k = 1..n), x, n+1), x, n), n=0..25); # _Muniru A Asiru_, Apr 23 2018

%t nmax = 25; CoefficientList[Series[Product[((1 + 8*x^k)/(1 - 8*x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 23 2018 *)

%t nmax = 30; CoefficientList[Series[(-7*QPochhammer[-8, x] / (9*QPochhammer[8, x]))^(1/8), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 23 2018 *)

%o (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+8*x^k)/(1-8*x^k))^(1/8)))

%Y Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), A303360 (b=2), this sequence (b=3).

%Y Cf. A303381.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 22 2018