A246814 Expansion of phi(-q) * phi(-q^4)^2 in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, -2, 8, 0, 0, -4, -10, 0, 0, 8, 8, 0, 0, 6, -16, 0, 0, -8, 16, 0, 0, -8, -10, 0, 0, 0, 24, 0, 0, 12, -16, 0, 0, -10, 8, 0, 0, -8, -32, 0, 0, 24, 24, 0, 0, 8, -18, 0, 0, -8, 24, 0, 0, -16, -16, 0, 0, 0, 24, 0, 0, 6, -32, 0, 0, -16, 32, 0, 0, -12

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

Euler transform of period 8 sequence [ -2, -1, -2, -5, -2, -1, -2, -3, ...].

a(n) = (-1)^(mod(n,4) = 1) * A116597(n).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A212885(n). a(4*n + 1) = -(-1)^n * A005876(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/16) * 2^(3/8) * (2+sqrt(2))^2 * Gamma(5/8)^3 / Gamma(7/8)^3 / Pi^(3/4) = A388862. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 - 2*q - 2*q^4 + 8*q^5 - 4*q^8 - 10*q^9 + 8*q^12 + 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; Table[a[n], {n, 0, 80}]

Prog

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

Crossrefs

Cf. A005876, A116597, A212885.

Sequence in context: A284455 A333250 A300222 * A116597 A202496 A165664

Adjacent sequences: A246811 A246812 A246813 * A246815 A246816 A246817

Keyword

sign

Author

Michael Somos, Sep 03 2014

Status

approved