A258228 Expansion of f(q) * f(-q^2) * chi(q^3) in powers of q where chi(), f() are Ramanujan theta functions.

1, 1, -2, 0, 1, -4, 0, 0, -2, 4, 2, 0, 0, 2, 0, 0, 1, -4, 4, 0, -4, 0, 0, 0, 0, 3, -4, 0, 0, -4, 0, 0, -2, 0, 2, 0, 4, 2, 0, 0, 2, -4, 0, 0, 0, 8, 0, 0, 0, 1, -6, 0, 2, -4, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, -8, 0, 0, -4, 0, 0, 0, 4, 2, -4, 0, 0, 0, 0, 0, -4, 4

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 f(q)^2 * f(-q^6) / f(q, q^5) in powers of q where f(,) is Ramanujan's general theta function.

Expansion of eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -4, 1, -2, 2, -3, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122865.

G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 * (1 + x^(3*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))).

a(n) = (-1)^n * A258210(n) = A258279(2*n) = A258292(2*n).

a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0.

a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(12*n + 1) = A002175(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * Pi^(1/6) * 3^(1/3) * Gamma(2/3)^(2/3) * Gamma(11/12)^(1/3) * Gamma(7/12) / (sqrt(2) * (1+3^(1/2)))^(1/3) * sqrt(2) * (1+3^(1/2)) / Gamma(3/4)^(10/3) = A388887. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + q - 2*q^2 + q^4 - 4*q^5 - 2*q^8 + 4*q^9 + 2*q^10 + 2*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^2 / (QPochhammer[ -q, q^6] QPochhammer[ -q^5, q^6]), {q, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] + A[2] - 2*A[3] + A[5] - 4*A[6] - 2*A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] - 4*A[18] + 4*A[19];

Crossrefs

Cf. A002175, A004018, A122856, A122865, A258210, A258279, A258292.

Sequence in context: A256276 A257920 A258210 * A271584 A072737 A061290

Adjacent sequences: A258225 A258226 A258227 * A258229 A258230 A258231

Keyword

sign

Author

Michael Somos, May 23 2015

Status

approved