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%I #35 May 06 2021 04:53:04
%S 1,-1,-2,-2,-5,-9,-36,-104,-365,-1219,-4213,-14617,-51570,-183084,
%T -656536,-2370066,-8613590,-31478538,-115632718,-426676244,
%U -1580878746,-5878933054,-21936060630,-82100980070,-308146839623,-1159545407027,-4373730398473,-16533813947503
%N Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.
%H Vaclav Kotesovec, <a href="/A309867/b309867.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ -c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 - 4*(1/4)^k))/2 = 0.4567034206737725013365271429022657551331606541289778092649... - _Vaclav Kotesovec_, May 06 2021
%t nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1-4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 06 2021 *)
%o (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1-4*x^k))/2))
%o (PARI) N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, binomial(2*j-2, j-1)*x^(i*j)/j)))
%Y Convolution inverse of A322204.
%Y Cf. A115140, A327682, A327683.
%K sign
%O 0,3
%A _Seiichi Manyama_, Sep 22 2019
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