A243747 Expansion of (phi(q) - phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

1, -2, 1, 2, -2, 0, 1, -2, 4, -2, -2, 2, 0, 0, 1, 0, -1, -2, 4, 0, -2, 0, -2, 2, 4, -4, 0, 2, 0, 0, 1, -4, 2, 0, -1, 2, -2, 0, 4, 0, 0, -2, -2, 2, 0, 0, -2, 0, 5, -4, 4, 2, -4, 0, 0, -4, 4, -2, 0, 2, 0, 0, 1, 4, -4, -2, 2, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0

Offset: 2

Author

Michael Somos, Jun 09 2014

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Expansion of third basis element of modular forms space for Gamma_1(8) of weight 1 in powers of q.

Examples

			G.f. = q^2 - 2*q^3 + q^4 + 2*q^5 - 2*q^6 + q^8 - 2*q^9 + 4*q^10 - 2*q^11 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

CF. A143259.

Programs

Magma

Basis( ModularForms( Gamma1(8), 1), 70) [3];

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];

PARI

{a(n) = if( n<2, 0, sum(k=1, n-1, (issquare(k) - issquare(2*k)) * (issquare(n - k) - issquare(2*n - 2*k))))};

Sage

ModularForms( Gamma1(8), 1, prec=70).2

Formula

Expansion of (q * f(-q, -q^7)^2 / psi(-q))^2 in powers of q where psi(), f() are Ramanujan theta functions.

Euler transform of period 8 sequence [-2, 0, 2, 2, 2, 0, -2, -2, ...].

G.f.: (theta_3(x) - theta_3(x^2))^2 / 4 = (Sum_{k>0} x^(k^2) - x^(2k^2))^2.

Convolution square of A143259.