A276847 Expansion of eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) in powers of q.

1, 0, -1, 0, -2, 0, 0, 0, 1, 0, 4, 0, -2, 0, 2, 0, 2, 0, -4, 0, 0, 0, -8, 0, -1, 0, -1, 0, 6, 0, 8, 0, -4, 0, 0, 0, 6, 0, 2, 0, -6, 0, 4, 0, -2, 0, 0, 0, -7, 0, -2, 0, -2, 0, -8, 0, 4, 0, 4, 0, -2, 0, 0, 0, 4, 0, -4, 0, 8, 0, 8, 0, 10, 0, 1, 0, 0, 0, -8, 0, 1, 0

Offset: 1

Author

Seiichi Manyama, Sep 22 2016

Keywords

• sign
• mult

Comments

The bisection of this sequence containing all nonzero terms is A030188.

Multiplicative. See A030188 for formula. - Andrew Howroyd, Jul 31 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Crossrefs

Cf. A030188, A271231, A276807, A276649.

Programs

Mathematica

CoefficientList[Series[QPochhammer[x^2] QPochhammer[x^4] QPochhammer[x^6] QPochhammer[x^12], {x, 0, 100}], x] (* Jan Mangaldan, Jan 04 2017 *)

Formula

a(4*n-3) = A271231(4*n-3), a(4*n-2) = 0, a(4*n-1) = -A271231(4*n-1), a(4*n) = 0.

G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(6*k)) * (1 - x^(12*k)).

a(2*n+1) = A030188(n). - Michel Marcus, Sep 25 2016

Euler transform of period 12 sequence [0, -1, 0, -2, 0, -2, 0, -2, 0, -1, 0, -4, ...]. - Georg Fischer, Nov 17 2022