A262056 Expansion of phi(q^2) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

1, 0, 2, 2, 0, 4, 4, 0, 10, 8, 0, 20, 14, 0, 36, 24, 0, 64, 42, 0, 108, 68, 0, 176, 108, 0, 280, 170, 0, 436, 260, 0, 666, 392, 0, 1000, 584, 0, 1480, 856, 0, 2160, 1240, 0, 3116, 1780, 0, 4448, 2526, 0, 6286, 3552, 0, 8804, 4956, 0, 12228, 6856, 0, 16852

Offset: 0

Author

Michael Somos, Sep 09 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*q^2 + 2*q^3 + 4*q^5 + 4*q^6 + 10*q^8 + 8*q^9 + 20*q^11 + ...
		

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. A143068.

Programs

Mathematica

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

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2), n))};

Formula

Expansion of eta(q^4)^5 * eta(q^6) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 24 sequence [ 0, 2, 2, -3, 0, 3, 0, -1, 2, 2, 0, -2, 0, 2, 2, -1, 0, 3, 0, -3, 2, 2, 0, 0, ...].

a(n) = A143068(2*n). a(3*n + 1) = 0.