A256004 Expansion of q^3 * f( -q, -q^8)^4 * f( -q^2, -q^7) / (f( -q) * f( -q^4, -q^5)^2) in powers of q where f() is Ramanujan's general theta function.

1, -3, 3, 0, 0, -3, 1, 3, 0, -2, -3, 3, 3, -3, 0, 0, 0, 0, 2, -3, 3, -3, 3, -3, 1, 0, 0, 3, -3, 0, 0, 0, 3, -2, 0, 0, -1, -3, 3, 3, 0, -6, 3, 3, 0, -2, -3, 6, 0, -6, 0, 0, 3, 0, 2, -3, 3, 0, 0, -3, 2, 0, 0, -3, -3, 0, 3, 6, 0, -3, 0, 0, 4, -6, 3, -3, 0, -3, 1

Offset

3,2

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 = 3..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 9 sequence [ -3, 0, 1, 3, 3, 1, 0, -3, -2, ...].

Example

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

Mathematica

a[ n_] := If[ n < 3, 0, With[{m = n - 3}, SeriesCoefficient[ q^3 Product[ (1 - q^k)^{3, 0, -1, -3, -3, -1, 0, 3, 2}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];

Prog

(PARI) {a(n) = if( n<3, 0, n-=3; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[2, 3, 0, -1, -3, -3, -1, 0, 3][k%9 + 1]), n))};
(Magma) Basis( ModularForms( Gamma1(9), 1), 82) [4];

Crossrefs

Sequence in context: A342697 A360480 A257094 * A269707 A109247 A021307

Adjacent sequences: A256001 A256002 A256003 * A256005 A256006 A256007

Keyword

sign

Author

Michael Somos, May 06 2015

Status

approved