A116597 Expansion of theta_3(q) * theta_4(q^4)^2 in powers of q.

1, 2, 0, 0, -2, -8, 0, 0, -4, 10, 0, 0, 8, -8, 0, 0, 6, 16, 0, 0, -8, -16, 0, 0, -8, 10, 0, 0, 0, -24, 0, 0, 12, 16, 0, 0, -10, -8, 0, 0, -8, 32, 0, 0, 24, -24, 0, 0, 8, 18, 0, 0, -8, -24, 0, 0, -16, 16, 0, 0, 0, -24, 0, 0, 6, 32, 0, 0, -16, -32, 0, 0, -12, 16, 0, 0, 24, -32, 0, 0, 24, 34, 0, 0, -16, -16, 0, 0, -8, 48

Offset: 0

Author

Michael Somos, Feb 18 2006

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 - 8*q^5 - 4*q^8 + 10*q^9 + 8*q^12 - 8*q^13 + 6*q^16 + 16*q^17 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A005876, A080963, A212885, A257536.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* Michael Somos, Apr 28 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * (eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)))^2, n))};

Formula

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

Expansion of eta(q^2)^5 * (eta(q^4) / (eta(q) * eta(q^8)))^2 in powers of q.

Euler transform of period 8 sequence [ 2, -3, 2, -5, 2, -3, 2, -3, ...].

G.f.: theta_3(q) * theta_4(q^4)^2 = Product_{k>0} (1 - x^(2*k))^3 *((1 + x^k) / (1 + x^(4*k)))^2.

a(4*n + 2) = a(4*n + 3) = 0. a(n) = A080963(4*n). a(4*n) = A212885(n). a(4*n + 1) = (-1)^n * A005876(n).

a(3*n + 1) = 2 * A257536(n). - Michael Somos, Apr 28 2015