A258771 Expansion of psi(-x) * phi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 7, 16, 7, -16, 0, 17, -48, -64, 16, 1, -16, 16, -32, 32, 55, -48, 64, 64, 16, 128, -9, -80, -32, 16, 48, -80, 96, 49, -144, -16, -144, -64, -64, -96, 144, 33, -64, -160, 0, 112, 32, 32, -96, 128, -25, 0, 32, -160, 304, 144, 96, 144, -48, 48, 119, 16, -256

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 q^(-1/8) * eta(q^2)^19 / (eta(q) * eta(q^4))^7 in powers of q.

Euler transform of period 4 sequence [ 7, -12, 7, -5, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1024 (t/i)^(5/2) f(t) where q = exp(2 Pi i t).

a(3*n + 2) = 16 * A258770(n).

Convolution square is A209942.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/2) * exp(Pi / 8) * Pi^(5/4) * 2^(1/4) / Gamma(3/4)^5 = A388894. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 7*x + 16*x^2 + 7*x^3 - 16*x^4 + 17*x^6 - 48*x^7 - 64*x^8 + ...
G.f. = q + 7*q^9 + 16*q^17 + 7*q^25 - 16*q^33 + 17*q^49 - 48*q^57 + ...

Mathematica

a[ n_] := SeriesCoefficient[EllipticTheta[ 3, 0, x]^4 QPochhammer[ x] / QPochhammer[ x^2, x^4], {x, 0, n}];

Prog

(PARI) {a(n) = my(A);  if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7, n))};

Crossrefs

Cf. A209942, A258770.

Sequence in context: A156377 A069526 A061039 * A063593 A070417 A101681

Adjacent sequences: A258768 A258769 A258770 * A258772 A258773 A258774

Keyword

sign

Author

Michael Somos, Jun 09 2015

Status

approved