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A303307 Expansion of Product_{n>=1} ((1 + (2*x)^n)/(1 - (2*x)^n))^(1/2).

%I A303307 #21 Oct 23 2018 10:33:39
%S A303307 1,2,6,20,54,156,444,1192,3174,8620,22516,58392,151996,387352,984888,
%T A303307 2507088,6270854,15659724,39067588,96454072,237663444,584266696,
%U A303307 1425921992,3470869296,8431325916,20380759544,49122457608,118178637040,283150466232,676768288176
%N A303307 Expansion of Product_{n>=1} ((1 + (2*x)^n)/(1 - (2*x)^n))^(1/2).
%H A303307 Seiichi Manyama, <a href="/A303307/b303307.txt">Table of n, a(n) for n = 0..3000</a>
%F A303307 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(k-1) * A054785(k) * a(n-k) for n > 0.
%F A303307 a(n) ~ 2^(n - 21/8) * exp(Pi*sqrt(n/2)) / n^(7/8). - _Vaclav Kotesovec_, Apr 21 2018
%t A303307 CoefficientList[Series[Sqrt[QPochhammer[-1, 2*x] / (2*QPochhammer[2*x])], {x, 0, 30}], x] (* _Vaclav Kotesovec_, Apr 21 2018 *)
%o A303307 (Ruby)
%o A303307 def s(n)
%o A303307   s = 0
%o A303307   (1..n).each{|i| s += i if n % i == 0}
%o A303307   s
%o A303307 end
%o A303307 def A303307(n)
%o A303307   ary = [1]
%o A303307   a = (0..n).map{|i| 2 ** (i - 1) * (s(2 * i) - s(i))}
%o A303307   (1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
%o A303307   ary
%o A303307 end
%o A303307 p A303307(100)
%Y A303307 Cf. A054785, A303306, A303342.
%K A303307 nonn
%O A303307 0,2
%A A303307 _Seiichi Manyama_, Apr 21 2018