A245669 Expansion of q * f(q, q^5)^3 in powers of q where f() is Ramanujan's two-variable theta function.

1, 3, 3, 1, 0, 3, 6, 3, 3, 6, 6, 3, 0, 6, 6, 1, 6, 9, 6, 0, 0, 12, 12, 3, 7, 6, 9, 6, 0, 12, 6, 3, 6, 6, 12, 3, 0, 12, 12, 6, 6, 12, 18, 6, 0, 12, 12, 3, 7, 15, 12, 0, 0, 9, 12, 6, 12, 18, 6, 6, 0, 18, 18, 1, 12, 12, 18, 6, 0, 12, 12, 9, 12, 18, 15, 6, 0, 18

Offset: 1

Author

Michael Somos, Jul 28 2014

Keywords

• nonn

Comments

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

Examples

			G.f. = q + 3*q^2 + 3*q^3 + q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + 6*q^10 + ...
		

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. A005885, A089801, A245668.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 3, 0, q^(1/3)] - EllipticTheta[ 3, 0, q^3]) / 2)^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)])^3 / (2^(3/2) q^(1/8)), {q, 0, n}];

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^3, n))};

Formula

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A245668.

G.f.: x * (Sum_{k in Z} x^(3*k^2 + 2*k))^3.

Convolution cube of A089801.

a(2*n + 2) = A005885(n).