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A303306 Expansion of Product_{n>=1} ((1 - (2*x)^n)/(1 + (2*x)^n))^(1/2).

%I A303306 #15 Oct 23 2018 10:32:42
%S A303306 1,-2,-2,-4,6,4,12,56,134,-108,196,328,-484,-88,-3752,-18576,16838,
%T A303306 -16460,-95340,-24408,-201036,-472584,565544,1424144,1843356,-6632568,
%U A303306 10365224,2317008,49620088,130484688,-4419664,631241440,761908550,-29690892,329427380,-8889717144,23673793860
%N A303306 Expansion of Product_{n>=1} ((1 - (2*x)^n)/(1 + (2*x)^n))^(1/2).
%H A303306 Seiichi Manyama, <a href="/A303306/b303306.txt">Table of n, a(n) for n = 0..3000</a>
%F A303306 a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} 2^(k-1) * A054785(k) * a(n-k) for n > 0.
%o A303306 (Ruby)
%o A303306 def s(n)
%o A303306   s = 0
%o A303306   (1..n).each{|i| s += i if n % i == 0}
%o A303306   s
%o A303306 end
%o A303306 def A303306(n)
%o A303306   ary = [1]
%o A303306   a = (0..n).map{|i| 2 ** (i - 1) * (s(2 * i) - s(i))}
%o A303306   (1..n).each{|i| ary << -(1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
%o A303306   ary
%o A303306 end
%o A303306 p A303306(100)
%Y A303306 Cf. A054785, A303307, A303343.
%K A303306 sign
%O A303306 0,2
%A A303306 _Seiichi Manyama_, Apr 21 2018