A080963 Expansion of theta_3(q)*theta_3(q^2)*theta_4(q^8) in powers of q.

1, 2, 2, 4, 2, 0, 4, 0, 0, 2, -4, -4, 0, 0, -8, 0, -2, -8, 6, -4, -8, 0, 4, 0, 0, -6, -12, 0, 0, 0, -8, 0, -4, 8, 8, -8, 10, 0, 12, 0, 0, 0, -8, 12, 0, 0, -8, 0, 8, 2, 14, 8, -8, 0, 16, 0, 0, 8, -4, 4, 0, 0, -16, 0, 6, 0, 16, -4, 16, 0, 8, 0, 0, 8, -20, -4, 0, 0, -8, 0, -8, -6, 8, 4, -16, 0, 20, 0, 0, -8, -20, -8, 0, 0, -16, 0, -8

Offset: 0

Author

Michael Somos, Feb 28 2003

Keywords

• sign

Comments

Ramanujan theta functions: f(q) := Product_{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

Programs

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]*EllipticTheta[3,0,q^2]* EllipticTheta[3,0,-q^8], {q,0,n}]; Table[a[n], {n,0,50}] (* or *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ (eta[q^2]*eta[q^4])^3/(eta[q]^2*eta[q^16]), {q,0,n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 11 2018 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^4+A))^3/(eta(x+A)^2*eta(x^16+A)), n))}

Formula

a(16*n+5) = a(16*n+7) = a(16*n+8) = a(16*n+12) = a(16*n+13) = a(16*n+15) = 0.

a(n) = 2*A080918(n) - A080917(n).

a(2*n+1) = 2*A034950(n).

Expansion of (eta(q^2)*eta(q^4))^3/(eta(q)^2*eta(q^16)) in powers of q.

Euler transform of period-16 sequence [2,-1,2,-4,2,-1,2,-4,2,-1,2,-4,2,-1,2,-3,...].

Expansion of phi(q)phi(q^2)phi(-q^8) in powers of q where phi() is a Ramanujan theta function.

G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))*(1-x^(4k))^2/((1+x^(4k))*(1+x^(8k))). - Michael Somos, Feb 16 2006