A260163 Expansion of f(x^2)^2 / f(-x) in powers of x where f() is a Ramanujan theta function.

1, 1, 4, 5, 8, 12, 17, 24, 36, 48, 65, 88, 116, 152, 200, 257, 328, 420, 532, 668, 840, 1045, 1296, 1604, 1972, 2416, 2952, 3592, 4357, 5272, 6356, 7640, 9168, 10964, 13080, 15576, 18497, 21920, 25932, 30604, 36048, 42392, 49752, 58288, 68184, 79617, 92820

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^(-1/8) * eta(q^4)^6 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.

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

a(n) = A132965(2*n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(Pi / 8) * 2^(3/4) * Gamma(5/8) * (2+sqrt(2)) / Pi^(1/4) / Gamma(7/8) = A388908. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + x + 4*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 17*x^6 + 24*x^7 + 36*x^8 + ...
G.f. = q + q^9 + 4*q^17 + 5*q^25 + 8*q^33 + 12*q^41 + 17*q^49 + 24*q^57 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^6 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))};

Crossrefs

Cf. A132965.

Sequence in context: A370428 A027975 A011980 * A362560 A061765 A242274

Adjacent sequences: A260160 A260161 A260162 * A260164 A260165 A260166

Keyword

nonn

Author

Michael Somos, Nov 09 2015

Status

approved