A030203 Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.

1, -1, -1, -1, 1, 2, -1, 2, 0, 0, -1, -1, -1, -1, 0, 1, -1, -1, 2, 0, 1, 2, 1, -1, 0, -1, 2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 2, 2, 0, -1, 1, 0, 1, 0, -1, 2, 2, 1, 0, -2, 2, -1, 0, -1, -1, -1, 1, -1, 0, 0, -1, -1, -1, 0, 0, 2, -2, -1, 0, -1, 1, 2, 2, 0, 0

Offset

0,6

Comments

Number 65 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x) * f(-x^3) where f(-x) := f(-x, -x^2) is a Ramanujan theta function. - Michael Somos, Jul 27 2006

Expansion of q^(-1/6) * sqrt(b(q) * c(q)/3) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006

Euler transform of period 3 sequence [ -1, -1, -2, ...]. - Michael Somos, Jul 27 2006

Given g.f. A(x), then B(q) = (q * A(q^6))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u^2*w - 4*u*w^2. - Michael Somos, Jul 27 2006

a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p = x^2 + 27*y^2, b(p^e) = [1, -1, 0] depending on e (mod 3) otherwise.

G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 22 2012

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)).

Example

G.f. = 1 - x - x^2 - x^3 + x^4 + 2*x^5 - x^6 + 2*x^7 - x^10 - x^11 - x^12 - x^13 + ...
G.f. = q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - q^67 - ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%6==5, (1 + (-1)^e) / 2, (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1)))}; /* Michael Somos, Jul 27 2006 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A), n))}; /* Michael Somos, Jul 27 2006 */
(Magma) Basis( CuspForms( Gamma1(108), 1), 452)[1]; /* Michael Somos, Jan 31 2015 */

Crossrefs

Sequence in context: A161491 A332998 A301652 * A208978 A333451 A101664

Adjacent sequences: A030200 A030201 A030202 * A030204 A030205 A030206

Keyword

sign

Author

N. J. A. Sloane.

Status

approved