A327682 Expansion of Product_{k>0} (-1+sqrt(1+4*x^k))/(2*x^k).

1, -1, 1, -5, 14, -40, 122, -404, 1362, -4608, 15881, -55709, 197402, -705114, 2539282, -9210196, 33605471, -123262137, 454268676, -1681305246, 6246544735, -23288217459, 87096982499, -326680267261, 1228547420236, -4631474743422, 17499462106763, -66257720483935, 251356773101419

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A184437 A023871 A274598 * A171185 A122485 A198086

Adjacent sequences: A327679 A327680 A327681 * A327683 A327684 A327685

Keyword

sign

Author

Seiichi Manyama, Sep 22 2019

Status

approved