A263767 Expansion of phi(-x) * psi(-x^8) * chi(x^24) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

1, -2, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 0, -4, 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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^2 * eta(q^8) * eta(q^32) * eta(q^48)^2 / (eta(q^2) * eta(q^16) * eta(q^24) * eta(q^96)) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 10368^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256574.

a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.

Example

G.f. = 1 - 2*x + 2*x^4 - x^8 - 2*x^12 + 2*x^16 + 2*x^17 - 2*x^24 - 2*x^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^24, x^48] EllipticTheta[ 2, Pi/4, x^4] EllipticTheta[ 4, 0, x] / (2^(1/2) x), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^2 + A) * eta(x^16 + A) * eta(x^24 + A) * eta(x^96 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^8)*eta(q^32)*eta(q^48)^2/(eta(q^2)*eta(q^16)* eta(q^24)*eta(q^96))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Cf. A256574.

Sequence in context: A258034 A243828 A034949 * A185338 A208603 A340988

Adjacent sequences: A263764 A263765 A263766 * A263768 A263769 A263770

Keyword

sign

Author

Michael Somos, Oct 25 2015

Status

approved