A224822 Expansion of phi(-q) * phi(-q^3)^2 in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, -4, 10, 0, 4, -16, 0, -2, 8, 0, 12, -8, 0, -16, 26, 0, 0, -24, 0, -8, 8, 0, 20, -10, 0, -4, 32, 0, 8, -48, 0, -8, 16, 0, 10, -8, 0, -32, 40, 0, 8, -24, 0, 0, 16, 0, 28, -18, 0, -24, 40, 0, 4, -64, 0, -8, 8, 0, 32, -24, 0, -16, 58, 0, 16, -24, 0, -16

Offset: 0

Author

Michael Somos, Jul 20 2013

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*q - 4*q^3 + 10*q^4 + 4*q^6 - 16*q^7 - 2*q^9 + 8*q^10 + 12*q^12 +
...
		

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. A028967, A224821.

Programs

Mathematica

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

PARI

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

Formula

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

Euler transform of period 6 sequence [-2, -1, -6, -1, -2, -3, ...].

G.f.: (Sum_{k in Z} (-1)^k * x^k^2) * (Sum_{k in Z} (-1)^k * x^(3*k^2))^2.

a(3*n + 2) = 0. a(2*n) = A028967(n). a(3*n) = A224821(n).