A107063 Expansion of q^(-1/24) * (eta(q^2) * eta(q^3)^4) / (eta(q) * eta(q^6)^2) in powers of q.

1, 1, 1, -2, -2, -1, 0, 1, -2, 0, -2, 0, 3, 2, 2, -1, 0, 2, -2, 2, 0, 0, 1, 0, 2, -2, 1, 0, -2, -4, 0, 0, -2, 0, 0, 1, 0, 0, 0, -2, 1, 0, -2, -2, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, -2, 2, 0, 1, 0, 0, 0, 0, -2, 4, 0, 0, 0, -2, 0, 2, 3, 0, 0, -2, 2, -2, -2, -1, -2, 0, -4, 0, 0, 2, -2, 0, 0, -2, 2, 2, -2, 0, 1, 0, 0, -2, 0, -4, 0, 2, 1, -2, 0, -2, 0

Offset: 0

Author

Michael Somos, May 10 2005

Keywords

• sign

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

A030204(3*n) = a(n).

Programs

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/24)* (eta[q^2]*eta[q^3]^4)/(eta[q]*eta[q^6]^2), {q, 0, 100}], q] (* G. C. Greubel, Apr 18 2018 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^4 / eta(x + A) / eta(x^6 + A)^2, n))}

Formula

Euler transform of period 6 sequence [1, 0, -3, 0, 1, -2, ...].

G.f.: Product_{k>0} (1+x^k)*(1-x^(3*k))^2/(1+x^(3*k))^2.

Expansion of phi(-q^3)^2 / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.