A246816 Expansion of phi(q) * phi(-q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

1, 2, -2, -4, 0, -4, 0, 8, -2, 6, 8, -4, 0, -12, 0, 8, -4, 8, -10, -12, 0, -8, 0, 8, 8, 14, 8, -16, 0, -4, 0, 16, 6, 16, -16, -8, 0, -20, 0, 8, -8, 8, 16, -20, 0, -20, 0, 16, -8, 18, -10, -8, 0, -12, 0, 24, 0, 16, 24, -12, 0, -20, 0, 24, 12, 8, -16, -28, 0

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)^7 / (eta(q)^2 * eta(q^4) * eta(q^8)) in powers of q.

Euler transform of period 8 sequence [ 2, -5, 2, -4, 2, -5, 2, -3, ...].

a(n) = (-1)^floor(n/2) * A127786(n). a(2*n) = A246814(n). a(2*n + 1) = 2 * A246815(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * 2^(5/16) * Gamma(5/8)^3 * (2+sqrt(2))^(3/2) / (2-sqrt(2))^(1/4) / Gamma(7/8)^3 / Pi^(3/4) = A388864. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 2*q - 2*q^2 - 4*q^3 - 4*q^5 + 8*q^7 - 2*q^8 + 6*q^9 + 8*q^10 + ...

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q^2]*EllipticTheta[3, 0, -q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)), n))};

Crossrefs

Cf. A127786, A246814, A246815.

Sequence in context: A063070 A049802 A129240 * A127786 A030207 A366561

Adjacent sequences: A246813 A246814 A246815 * A246817 A246818 A246819

Keyword

sign

Author

Michael Somos, Sep 03 2014

Status

approved