A121456 Expansion of q*(psi(-q)*psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function.

1, -2, 1, -4, 6, -2, 8, -8, 1, -12, 12, -4, 14, -16, 6, -16, 18, -2, 20, -24, 8, -24, 24, -8, 31, -28, 1, -32, 30, -12, 32, -32, 12, -36, 48, -4, 38, -40, 14, -48, 42, -16, 44, -48, 6, -48, 48, -16, 57, -62, 18, -56, 54, -2, 72, -64, 20, -60, 60, -24, 62, -64, 8, -64, 84, -24, 68, -72, 24, -96, 72, -8, 74, -76, 31, -80

Offset: 1

Author

Michael Somos, Jul 30 2006

Keywords

• sign
• easy
• mult

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) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q];  a[n_]:= SeriesCoefficient[(eta[q] *eta[q^3]*eta[q^4]*eta[q^12])^2/(eta[q^2]*eta[q^6])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)

PARI

{a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,(n/d%2)*d*(d%3>0)))}

Formula

Expansion of (eta(q)*eta(q^3)*eta(q^4)*eta(q^12))^2/(eta(q^2)*eta(q^6))^2 in powers of q.

Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 0, -2, -2, -4, 0, -2, -4, ...].

Multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.

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

a(n) = (-1)^(n+1)*A111932(n).

Dirichlet g.f.: (1 - 5/2^s + 1/2^(2*s-2) ) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023