A145728 Expansion of f(-q^6) * f(-q^10) / (f(q) * f(q^15)) in powers of q where f() is a Ramanujan theta function.

1, -1, 2, -3, 5, -7, 10, -14, 20, -27, 36, -48, 63, -82, 106, -137, 175, -222, 280, -352, 439, -546, 676, -834, 1024, -1253, 1528, -1857, 2250, -2718, 3276, -3936, 4718, -5640, 6728, -8006, 9507, -11266, 13324, -15726, 18526, -21786, 25574, -29970, 35064

Offset

0,3

Comments

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

Essentially a duplicate of A094023. - N. J. A. Sloane, Nov 04 2008

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^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60)) / (eta(q^2) * eta(q^30))^3 in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 05 2015

a(n) = (-1)^n * A094023(n). Convolution inverse of A145727.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = 1 - q + 2*q^2 - 3*q^3 + 5*q^4 - 7*q^5 + 10*q^6 - 14*q^7 + 20*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^6] QPochhammer[ q^10] / (QPochhammer[ -q] QPochhammer[ -q^15]), {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A)) / (eta(x^2 + A) * eta(x^30 + A))^3, n))};

Crossrefs

Cf. A094023, A145727.

Sequence in context: A288254 A023893 A065094 * A145786 A094023 A123630

Adjacent sequences: A145725 A145726 A145727 * A145729 A145730 A145731

Keyword

sign

Author

Michael Somos, Oct 23 2008

Status

approved