A242609 Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, 2, 0, 0, 0, 2, -6, 0, 0, 4, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 2, -8, 0, 0, 6, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, -4, 0, 0, 4, 0, 0, 0, 0, -10, 0

Offset: 0

Author

Michael Somos, May 19 2014

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 + 2*q^4 + 2*q^8 - 6*q^9 + 4*q^12 + 2*q^16 - 4*q^17 + ...
		

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. A033715, A112603, A113411, A226225.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

PARI

{a(n) = if( n<1, n==0, 2 * (-1)^n * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^16 + A)^5 / (eta(x^2 + A) * eta(x^8 + A)^2 * eta(x^32 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^16)^5 / (eta(q^2) * eta(q^8)^2 * eta(q^32)^2) in powers of q.

G.f.: (Sum_{k in Z} (-x)^k^2) * (Sum_{k in Z} (x^8)^k^2).

a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0. a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = -2 * A112603(n). a(8*n + 4) = 2 * A113411(n).

a(n) = (-1)^n * A226225(n).