A309867 Expansion of Product_{k>0} (1+sqrt(1-4*x^k))/2.

1, -1, -2, -2, -5, -9, -36, -104, -365, -1219, -4213, -14617, -51570, -183084, -656536, -2370066, -8613590, -31478538, -115632718, -426676244, -1580878746, -5878933054, -21936060630, -82100980070, -308146839623, -1159545407027, -4373730398473, -16533813947503

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

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

Mathematica

nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1-4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2021 *)

Prog

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

Crossrefs

Convolution inverse of A322204.

Cf. A115140, A327682, A327683.

Sequence in context: A003228 A184713 A110182 * A304584 A193899 A334017

Adjacent sequences: A309864 A309865 A309866 * A309868 A309869 A309870

Keyword

sign

Author

Seiichi Manyama, Sep 22 2019

Status

approved