A145783 Expansion of (chi(q) * chi(q^15)) / (chi(q^3) * chi(q^5))^2 in powers of q where chi() is a Ramanujan theta function.

1, 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

0,8

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^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30))^2 / (eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^10)^4 * eta(q^15) * eta(q^60)) 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) = A145785(n) unless n=0.

Convolution inverse of A145782.

Example

G.f. = 1 + 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[ QPochhammer[ -x, x^2] QPochhammer[ -x^15, x^30] / (QPochhammer[ -x^3, x^6] QPochhammer[ -x^5, x^10] )^2 , {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^5 + A)^2 * eta(x^12 + A)^2 * eta(x^20 + A)^2 * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^10 + A)^4 * eta(x^15 + A) * eta(x^60 + A)), n))};

Crossrefs

Cf. A145782, A145785.

Sequence in context: A043754 A144191 A288007 * A145785 A094022 A134177

Adjacent sequences: A145780 A145781 A145782 * A145784 A145785 A145786

Keyword

sign

Author

Michael Somos, Oct 23 2008

Status

approved