A246608 Expansion of phi(-q) * phi(-q^4)^4 in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, -6, 16, 0, 0, 8, -50, 0, 0, 16, 80, 0, 0, -38, -96, 0, 0, -16, 160, 0, 0, 48, -242, 0, 0, 64, 240, 0, 0, -56, -288, 0, 0, -150, 400, 0, 0, 112, -384, 0, 0, 112, 496, 0, 0, -112, -674, 0, 0, -80, 560, 0, 0, 160, -672, 0, 0, 192, 880, 0, 0, -294

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

a(4*n) = A245643(n). a(4*n + 1) = -2 * A244276(n). a(4*n + 2) = a(4*n + 3) = 0.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = -(1/16) * (3/2+sqrt(2)) * 2^(3/4) * (2+sqrt(2))^(1/2) * Gamma(5/8)^5 / Pi^(5/4) / Gamma(7/8)^5 / (sqrt(2)-2) = A388836. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 - 2*q - 6*q^4 + 16*q^5 + 8*q^8 - 50*q^9 + 16*q^12 + 80*q^13 + ...

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]*EllipticTheta[3, 0, -q^4 ]^4, {q, 0, n}]; (* corrected by G. C. Greubel, Mar 15 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^8 / (eta(x^2 + A) * eta(x^8 + A)^4), n))};
(Magma) A := Basis( ModularForms( Gamma0(8), 5/2), 68); A[1] - 2*A[2];

Crossrefs

Cf. A244276, A245643.

Sequence in context: A348639 A244142 A161800 * A100344 A370796 A094596

Adjacent sequences: A246605 A246606 A246607 * A246609 A246610 A246611

Keyword

sign

Author

Michael Somos, Sep 01 2014

Status

approved