A327682 - OEIS

%I #34 May 06 2021 05:03:36

%S 1,-1,1,-5,14,-40,122,-404,1362,-4608,15881,-55709,197402,-705114,

%T 2539282,-9210196,33605471,-123262137,454268676,-1681305246,

%U 6246544735,-23288217459,87096982499,-326680267261,1228547420236,-4631474743422,17499462106763,-66257720483935,251356773101419

%N Expansion of Product_{k>0} (-1+sqrt(1+4*x^k))/(2*x^k).

%H Seiichi Manyama, <a href="/A327682/b327682.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ (-1)^n * c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (-1 + sqrt(1 + 4*(-1/4)^k)) / (2*(-1/4)^k) = 0.5396673413761086071059510679780476790558662471136055... - _Vaclav Kotesovec_, May 06 2021

%t m = 28; CoefficientList[Series[Product[(-1 + Sqrt[1 + 4*x^k])/(2*x^k), {k, 1, m}], {x, 0, m}], x] (* _Amiram Eldar_, May 06 2021 *)

%o (PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (-1+sqrt(1+4*x^k))/(2*x^k)))

%o (PARI) N=66; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, N\i, (-1)^j*binomial(2*j, j)*x^(i*j)/(j+1))))

%Y Cf. A000108, A081362, A168491, A309867, A322204, A327683.

%K sign

%O 0,4

%A _Seiichi Manyama_, Sep 22 2019