A138518 Expansion of (phi(-q) / phi(-q^5))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 4, 0, 4, -4, -16, 16, 4, 12, -12, -48, 48, 8, 32, -32, -124, 120, 20, 80, -76, -288, 272, 48, 176, -164, -616, 576, 96, 360, -336, -1248, 1156, 192, 712, -656, -2412, 2216, 368, 1344, -1228, -4488, 4096, 672, 2448, -2228, -8096, 7344, 1200, 4348, -3932

Offset

0,2

Comments

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

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

Formula

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

Euler transform of period 10 sequence [ -4, -2, -4, -2, 0, -2, -4, -2, -4, 0, ...].

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u - 5) * v * (v - 1) * (v - 5).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138519.

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

a(n) = -4 * A095813(n) unless n=0.

Convolution inverse of A138517. Convolution square of A138527.

Example

G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 4*q^5 - 16*q^6 + 16*q^7 + 4*q^8 + 12*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5])^2, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *)

Prog

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

Crossrefs

Cf. A095813, A138517, A138519, A138527.

Sequence in context: A104287 A174611 A283361 * A290799 A155836 A337398

Adjacent sequences: A138515 A138516 A138517 * A138519 A138520 A138521

Keyword

sign

Author

Michael Somos, Mar 23 2008

Status

approved