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%I #19 Apr 23 2018 18:36:42
%S 1,2,0,10,-10,38,-76,310,-960,3190,-10672,37262,-130170,459690,
%T -1639940,5901498,-21376154,77900710,-285457200,1051118590,
%U -3887169486,14431323506,-53766825940,200964040290,-753348868380,2831669141514,-10670007388128
%N Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).
%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = -4.
%H Seiichi Manyama, <a href="/A303350/b303350.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ -(-1)^n * sqrt(c) * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=2} (1 + 4*(-1/4)^k) = 1.1864623436704848646891654544376222586... - _Vaclav Kotesovec_, Apr 22 2018
%p seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # _Muniru A Asiru_, Apr 22 2018
%t nmax = 30; CoefficientList[Series[Product[(1 + 4*x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 22 2018 *)
%o (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+4*x^k)^(1/2)))
%Y Expansion of Product_{n>=1} (1 + b^2*x^n)^(1/b): A000009 (b=1), this sequence (b=2), A303351 (b=3).
%Y Cf. A261568, A303347, A303352.
%K sign
%O 0,2
%A _Seiichi Manyama_, Apr 22 2018
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