A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024

Offset: 0

Author

William J. Keith, Jan 18 2018

Keywords

• sign
• easy

Comments

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.

Sequence appears to be positive and negative roughly half the time.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Links

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

Crossrefs

Cf. A000108, A271235, A298994.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Programs

Mathematica

Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]

PARI

q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

Formula

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).