A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

1, 0, 1, -3, 0, 0, 2, 0, 1, 0, 0, -3, 2, 0, 0, -3, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, -6, 0, 0, 2, 0, 0, 0, 0, -3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, -3, 3, 0, 0, -6, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, -6, 0, 0, 2, 0, 1, 0, 0, -6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0, 0

Offset: 1

Author

Michael Somos, Feb 09 2006

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A033716, A096936, A115978.

Programs

Maple

S:=series((1-JacobiTheta4(0,q)*JacobiTheta4(0,q^3))/2, q, 106):
seq(coeff(S,q,n),n=1..105); # Robert Israel, Nov 20 2017

Mathematica

Drop[CoefficientList[Series[(1 -EllipticTheta[4, 0, q]*EllipticTheta[4, 0, q^3])/2, {q, 0, 110}], q], 1] (* G. C. Greubel, May 09 2019 *)

PARI

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^3+A))^2/eta(x^2+A)/eta(x^6+A), n)/-2)}

Sage

def E(x): return 1 + 2*sum((-1)^k*x^(k^2) for k in (1..50))
a=((1 - E(x)*E(x^3))/2).series(x, 110).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 09 2019

Scheme

(define (A115979 n) (- (* (expt -1 n) (A096936 n)))) ;; Follow A096936 for the rest of code. - Antti Karttunen, Nov 20 2017

Formula

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

Moebius transform is period 12 sequence [1,-1,0,-3,-1,0,1,3,0,1,-1,0,...].

a(n) is multiplicative and a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e)=1, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

G.f.: Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k)).

a(n) = -(-1)^n*A096936(n).

A115978(n) = -2*a(n) if n > 0.