A145724 Expansion of q * f(-q^20) / (f(q) * chi(-q^5)) in powers of q where f(), chi() are Ramanujan theta functions.

1, -1, 2, -3, 5, -6, 10, -13, 19, -25, 36, -46, 64, -82, 110, -139, 184, -231, 300, -375, 480, -596, 754, -930, 1165, -1428, 1772, -2162, 2662, -3230, 3952, -4773, 5800, -6976, 8430, -10093, 12136, -14476, 17320, -20585, 24526, -29044, 34466, -40684, 48095

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..45.

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^10) * eta(q^20) / (eta(q^2)^3 * eta(q^5)) in powers of q.

Euler transform of period 20 sequence [ -1, 2, -1, 1, 0, 2, -1, 1, -1, 2, -1, 1, -1, 2, 0, 1, -1, 2, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147701.

Convolution inverse of A145723.

a(n) ~ -(-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/5)) / (2^(5/2) * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 6*q^6 + 10*q^7 - 13*q^8 + 19*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^20] QPochhammer[ -q^5, q^5] / QPochhammer[ -q], {q, 0, n}]; (* Michael Somos, Sep 05 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^10 + A) * eta(x^20 + A) / (eta(x^2 + A)^3 * eta(x^5 + A)), n))};

Crossrefs

Cf. A145723, A147701.

Sequence in context: A039837 A039838 A064173 * A039843 A305937 A286097

Adjacent sequences: A145721 A145722 A145723 * A145725 A145726 A145727

Keyword

sign

Author

Michael Somos, Nov 10 2008

Status

approved