A030218 Expansion of eta(q^3) * eta(q^5) * eta(q^6) * eta(q^10) in powers of q.

0, 1, 0, 0, -1, 0, -1, -2, 0, 1, 1, -2, 2, 0, 2, -1, 3, 4, 0, 0, -2, -2, -2, 0, -1, 1, -2, 0, -2, -4, 0, 4, -4, 2, -2, 2, -1, -4, 4, 2, 1, 2, 2, 0, 6, 0, 0, 4, -2, 1, 0, 2, 4, -8, -1, -2, 2, -4, 2, -2, 1, -6, -4, -2, -1, -2, 2, 4, 0, 0, -2, -4, 0, 6, -6, 0, -8, 0, 0, 4

Offset

0,8

Links

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

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

Formula

Expansion of q * f(-q^3) * f(-q^5) * f(-q^6) * f(-q^10) in powers of q where f() is a Ramanujan theta function. - Michael Somos, Nov 17 2014

Euler transform of period 30 sequence [0, 0, -1, 0, -1, -2, 0, 0, -1, -2, 0, -2, 0, 0, -2, 0, 0, -2, 0, -2, -1, 0, 0, -2, -1, 0, -1, 0, 0, -4, ...]. - Michael Somos, Nov 17 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 30 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 17 2014

G.f. = x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(6*k)) * (1 - x^10*k)). - Michael Somos, Nov 17 2014

a(n) = -A286137(3*n). - Michael Somos, Mar 10 2020

Example

G.f. = q - q^4 - q^6 - 2*q^7 + q^9 + q^10 - 2*q^11 + 2*q^12 + 2*q^14 - q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ q^6] QPochhammer[ q^10], {q, 0, n}]; (* Michael Somos, Nov 17 2014 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^10 + A), n))}; /* Michael Somos, Nov 17 2014 */
(Magma) Basis( CuspForms( Gamma0(30), 2), 80) [1]; /* Michael Somos, Apr 27 2015 */

Crossrefs

Cf. A286137.

Sequence in context: A007968 A236532 A077763 * A281388 A127440 A118198

Adjacent sequences: A030215 A030216 A030217 * A030219 A030220 A030221

Keyword

sign

Author

N. J. A. Sloane.

Status

approved