A254346 Expansion of f(x, x^5) * f(-x^6) / f(x)^2 in powers of x where f() is a Ramanujan theta function.

1, -1, 3, -5, 10, -15, 26, -39, 63, -92, 140, -201, 295, -415, 591, -818, 1140, -1554, 2126, -2861, 3855, -5126, 6816, -8970, 11793, -15372, 20007, -25857, 33356, -42771, 54734, -69683, 88530, -111968, 141312, -177642, 222842, -278557, 347484, -432095, 536230

Offset

0,3

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^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^2)^4 in powers of q.

Euler transform of period 12 sequence [ -1, 3, -2, 2, -1, 2, -1, 2, -2, 3, -1, 0, ...].

a(n) = (-1)^n * A132302(n). 2 * a(n) = A254372(2*n + 1).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/128) * exp(Pi / 2) * 2^(1/3) * 3^(2/3) * (1+3^(1/2))^5 * Gamma(2/3)^(4/3) * Gamma(11/12)^(11/3) * Gamma(7/12)^5 * (11*3^(1/2)-19) / Gamma(3/4)^(26/3) / Pi^(2/3) = A388877. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 - x + 3*x^2 - 5*x^3 + 10*x^4 - 15*x^5 + 26*x^6 - 39*x^7 + ...
G.f. = q - q^3 + 3*q^5 - 5*q^7 + 10*q^9 - 15*q^11 + 26*q^13 - 39*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^12] / (QPochhammer[ x^2] QPochhammer[ -x]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^2 + A)^4, n))};

Crossrefs

Cf. A132302, A254372.

Sequence in context: A070557 A225751 A264397 * A132302 A308872 A097513

Adjacent sequences: A254343 A254344 A254345 * A254347 A254348 A254349

Keyword

sign

Author

Michael Somos, Jan 29 2015

Status

approved