A245668 Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

1, -3, 3, -1, -3, 6, -3, 0, 3, 3, -12, 6, -1, -12, 12, 0, -3, 12, 9, -12, 6, -6, -12, 0, -3, -15, 18, 5, 0, 18, -6, 0, 3, -6, -24, 12, 3, -12, 18, 0, -12, 24, -6, -12, 6, 18, -24, 0, -1, -27, 21, -6, -12, 18, 15, 0, 12, -6, -12, 18, 0, -36, 24, 0, -3, 24, -12

Offset: 0

Author

Michael Somos, Jul 28 2014

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A089807, A213592, A213607, A245669.

Programs

Magma

A := Basis( ModularForms( Gamma0(12), 3/2), 67);  A[1] - 3*A[2] + 3*A[3];

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q,0,n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q,0,n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};

Formula

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

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

Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.

a(3*n + 1) = -3 * A213056(n). a(6*n + 2) = 3 * A213592(n). a(6*n + 5) = 6 * A213607(n). a(8*n + 7) = 0.

Convolution cube of A089807.