A303124 Expansion of Product_{n>=1} (1 + (16*x)^n)^(1/4).

1, 4, 40, 1504, 10336, 387968, 5349632, 111442944, 1100563968, 36711258112, 493805416448, 9186633203712, 134635599806464, 2648342619422720, 43443234834350080, 938422838970810368, 11378951438668791808, 224791017150689574912, 4129154423023897411584

Offset: 0

Author

Seiichi Manyama, Apr 19 2018

Keywords

• nonn

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/4, g(n) = -16^n.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(1/b): A000009 (b=1), A298994 (b=2), A303074 (b=3), this sequence (b=4), A303125 (b=5).

Cf. A303131, A303153.

Programs

Mathematica

CoefficientList[Series[(QPochhammer[-1, 16*x]/2)^(1/4), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)

PARI

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(16*x)^k)^(1/4)))

Formula

a(n) ~ 2^(4*n - 17/8) * exp(sqrt(n/3)*Pi/2) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018