A145785 Expansion of q * f(-q^2) * f(-q^30) / (f(q^3) * f(q^5)) in powers of q where f() is a Ramanujan theta function.

1, 0, -1, -1, -1, 0, 2, 2, -1, -2, 0, 1, 2, 2, -3, -7, -2, 6, 8, 5, -2, -12, -10, 6, 13, 4, -7, -14, -10, 14, 32, 12, -24, -36, -22, 13, 50, 36, -26, -56, -22, 30, 62, 40, -51, -114, -46, 79, 129, 76, -54, -170, -114, 90, 192, 82, -104, -216, -132, 159, 350

Offset

1,7

Comments

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

Seems to differ from A145783 only by not defining a(0) (b-file heuristics for n<=1000). - R. J. Mathar, Mar 25 2024

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 eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30) / (eta(q^6) * eta(q^10))^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).

a(n) = -(-1)^n * A094022(n). Convolution inverse of A058727.

Example

G.f. = q - q^3 - q^4 - q^5 + 2*q^7 + 2*q^8 - q^9 - 2*q^10 + q^12 + 2*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2] QPochhammer[ q^30] / (QPochhammer[ -q^3] QPochhammer[ -q^5]), {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A) / (eta(x^6 + A) * eta(x^10 + A))^3, n))};

Crossrefs

Cf. A058727, A094022.

Sequence in context: A144191 A288007 A145783 * A094022 A134177 A190615

Adjacent sequences: A145782 A145783 A145784 * A145786 A145787 A145788

Keyword

sign

Author

Michael Somos, Oct 23 2008

Status

approved