A303350 Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).

1, 2, 0, 10, -10, 38, -76, 310, -960, 3190, -10672, 37262, -130170, 459690, -1639940, 5901498, -21376154, 77900710, -285457200, 1051118590, -3887169486, 14431323506, -53766825940, 200964040290, -753348868380, 2831669141514, -10670007388128

Offset: 0

Author

Seiichi Manyama, Apr 22 2018

Keywords

• sign

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = -4.

Links

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

Crossrefs

Expansion of Product_{n>=1} (1 + b^2*x^n)^(1/b): A000009 (b=1), this sequence (b=2), A303351 (b=3).

Cf. A261568, A303347, A303352.

Programs

Maple

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

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 + 4*x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

PARI

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

Formula

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