A133637 Expansion of q^(-1) * psi(-q) / psi(-q^3)^3 in powers of q where psi() is a Ramanujan theta function.

1, -1, 0, 2, -3, 0, 4, -6, 0, 10, -12, 0, 20, -24, 0, 36, -45, 0, 64, -78, 0, 112, -132, 0, 189, -222, 0, 308, -363, 0, 492, -576, 0, 778, -900, 0, 1210, -1392, 0, 1844, -2121, 0, 2776, -3180, 0, 4144, -4716, 0, 6114, -6936, 0, 8914, -10098, 0, 12884, -14550

Offset

-1,4

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (3 * c(q^2)) / (c(q) * c(q^4)) in powers of q where c() is a cubic AGM function.

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (3/4)^(1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132974.

a(3*n + 1) = 0. a(3*n) = - A132974(n).

Convolution inverse of A113427.

a(3*n - 1) = A263993(n). - Michael Somos, Oct 31 2015

Example

G.f. = 1/q - 1 + 2*q^2 - 3*q^3 + 4*q^5 - 6*q^6 + 10*q^8 - 12*q^9 + 20*q^11 - ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A113427, A132974, A263993.

Sequence in context: A271439 A334655 A343410 * A258093 A286578 A010340

Adjacent sequences: A133634 A133635 A133636 * A133638 A133639 A133640

Keyword

sign

Author

Michael Somos, Sep 18 2007

Status

approved