A263502 Expansion of phi(q) * f(-q^2)^3 / f(-q^6) in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, -3, -6, 2, 0, 0, 12, -3, -4, 12, -6, -6, 0, -6, 0, 2, -6, -12, 12, 0, 0, 24, -12, 0, 14, -6, -6, 12, 0, -24, 12, -3, 0, 12, -12, -4, 0, -12, -24, 12, -6, 0, 36, -6, 0, 24, -12, -6, 14, -15, 0, 0, 0, 0, 24, -6, -24, 36, -6, 0, 0, -18, -24, 2, -12, -24, 36

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36* t)) = 17496^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261444.

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

Euler transform of period 12 sequence [2, -6, 2, -4, 2, -5, 2, -4, 2, -6, 2, -3, ...].

a(n) = A263456(4*n). a(8*n + 5) = a(9*n + 6) = 0.

a(3*n + 2) = -3 * A261444(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * Pi^(7/12) * 3^(1/6) * sqrt(2) * (3^(1/2)-1) * Gamma(2/3)^(1/3) * (sqrt(2) * (1+3^(1/2)))^(4/3) / Gamma(11/12)^(4/3) / Gamma(3/4)^(2/3) / Gamma(7/12) = A388940. - Simon Plouffe, Sep 21 2025

Example

G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 + 12*x^7 - 3*x^8 - 4*x^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] QPochhammer[ q^2]^3 / QPochhammer[ q^6], {q, 0, n}];

Prog

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

Crossrefs

Cf. A261444, A263456.

Sequence in context: A084459 A093095 A260611 * A002171 A138515 A107410

Adjacent sequences: A263499 A263500 A263501 * A263503 A263504 A263505

Keyword

sign

Author

Michael Somos, Oct 19 2015

Status

approved