A279955 Expansion of chi(-x^4)^4 * f(-x^4)^2 * f(-x)^2 in powers of x where chi(), f() are Ramanujan theta functions.

1, -2, -1, 2, -5, 14, 4, -12, 5, -40, 0, 26, 11, 68, -15, -30, -18, -106, 3, 50, -10, 182, 29, -104, 10, -270, 11, 130, 37, 360, -51, -164, -16, -506, -30, 266, -65, 686, 62, -320, 53, -898, 22, 414, 50, 1206, -61, -612, -52, -1560, -4, 696, -81, 1958, 120

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

Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * eta(q^4)^2 * eta(q^16)^6 / eta(q^32)^4 in powers of q^4.

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

a(n) = (-1)^n * A280339(n).

a(3*n + 1) / a(1) == A002171(n) (mod 3). a(3^3*n + 7) / a(7) == A002171(n) (mod 3^2).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = -(1/8) * exp(-Pi / 4) * 2^(3/4) * Gamma(5/8)^4 * (3+2 * sqrt(2)) / (sqrt(2)-2) / Pi / Gamma(7/8)^4 = A388964. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 - 2*x - x^2 + 2*x^3 - 5*x^4 + 14*x^5 + 4*x^6 - 12*x^7 + 5*x^8 + ...
G.f. = q^-1 - 2*q^3 - q^7 + 2*q^11 - 5*q^15 + 14*q^19 + 4*q^23 - 12*q^27 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^4]^2 QPochhammer[ x^4, x^8]^4, {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^4 + A)^6 / eta(x^8 + A)^4, n))};

Crossrefs

Cf. A002171, A280339.

Sequence in context: A327194 A160457 A107087 * A280339 A115141 A031148

Adjacent sequences: A279952 A279953 A279954 * A279956 A279957 A279958

Keyword

sign

Author

Michael Somos, Dec 23 2016

Status

approved