A139136 Expansion of psi(-q) / f(q^3) where psi(), f() are Ramanujan theta functions.

1, -1, 0, -2, 1, 0, 4, -2, 0, -6, 4, 0, 10, -6, 0, -16, 9, 0, 24, -14, 0, -36, 20, 0, 52, -29, 0, -74, 42, 0, 104, -58, 0, -144, 80, 0, 198, -110, 0, -268, 148, 0, 360, -198, 0, -480, 264, 0, 634, -347, 0, -832, 454, 0, 1084, -592, 0, -1404, 764, 0, 1808, -982

Offset

0,4

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139135.

G.f.: Product_{k>0} P(12, x^k) / ( (1 + x^(2*k-1))^2 * P(3, x^k) * P(6, x^k)^2) where P(n, x) is n-th cyclotomic polynomial.

a(3*n) = A132002(n). a(3*n + 1) = - A139135(n). a(3*n + 2) = 0.

a(n) = (-1)^n * A122792(n). - Michael Somos, Sep 07 2015

Example

G.f. = 1 - q - 2*q^3 + q^4 + 4*q^6 - 2*q^7 - 6*q^9 + 4*q^10 + 10*q^12 - 6*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2^(-1/2) q^(-1/8) EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

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) * eta(x^6 + A)^3), n))};

Crossrefs

Cf. A122792, A132002, A139135.

Sequence in context: A122073 A106236 A270640 * A122792 A348218 A138002

Adjacent sequences: A139133 A139134 A139135 * A139137 A139138 A139139

Keyword

sign

Author

Michael Somos, Apr 10 2008

Status

approved