A261968 Expansion of phi(q^5) / phi(q) in powers of q where phi() is a Ramanujan theta function.

1, -2, 4, -8, 14, -22, 36, -56, 84, -126, 184, -264, 376, -528, 732, -1008, 1374, -1856, 2492, -3320, 4394, -5784, 7568, -9848, 12756, -16442, 21096, -26960, 34312, -43500, 54956, -69184, 86804, -108576, 135392, -168336, 208722, -258096, 318320, -391632

Offset: 0

Author

Michael Somos, Sep 06 2015

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^2 - 8*q^3 + 14*q^4 - 22*q^5 + 36*q^6 - 56*q^7 + ...
		

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. A138526, A144377.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q], {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5 / (eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^5 / (eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2) in powers of q.

Euler transform of period 20 sequence [ -2, 3, -2, 1, 0, 3, -2, 1, -2, 0, -2, 1, -2, 3, 0, 1, -2, 3, -2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - 2*u^2 + 5*u^4) * (1 - 2*v^2 + 5*v^4) - 4*(u^2 + 2*u*v - v^2)^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 + 3*u*v - u^2) * (u^2 + v^2) - u*v * (1 + 5*u^2*v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144377.

G.f.: Product_{k>0} P(10, x^k)^3 * P(5, x^k) / P(20, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.

a(n) = (-1)^n * A138526(n). Convolution inverse is A144377.