A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

1, -3, 0, 2, 9, 0, -22, 0, 0, 26, -6, 0, 25, -27, 0, -46, 0, 0, 26, 66, 0, -22, 0, 0, -45, 0, 0, 0, -78, 0, 74, 18, 0, 122, 0, 0, -46, -75, 0, -142, 81, 0, 0, 0, 0, -44, 138, 0, 2, 0, 0, 194, 0, 0, -214, -78, 0, 0, -198, 0, 121, 0, 0, 146, 66, 0, 52, 0, 0, -22

Offset

0,2

Comments

Number 28 of the 74 eta-quotients listed in Table I of Martin (1996).

Expansion of a newform level 12 weight 3 and character [0, 1].

Links

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

Author?, Eta Products and Quotients which are Newforms. [Broken link?]

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

Formula

Euler transform of period 3 sequence [ -3, -3, -6, ...]. - Michael Somos, Feb 13 2006

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-3)^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006

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

G.f.: Sum_{k>=0} a(k) * q^(2*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * q^(u*u + 3*v*v). - Michael Somos, Jun 14 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008

a(3*n + 1) = -3 * a(n). a(3*n + 2) = 0. a(3*n) = A152243(n). - Michael Somos, Mar 09 2012

a(n) = (-1)^n * A209939(n). - Michael Somos, Mar 16 2012

Convolution square is A007332. - Michael Somos, Nov 16 2008

Example

G.f. = 1 - 3*x + 2*x^3 + 9*x^4 - 22*x^6 + 26*x^9 - 6*x^10 + 25*x^12 - 27*x^13 + ...
G.f. = q - 3*q^3 + 2*q^7 + 9*q^9 - 22*q^13 + 26*q^19 - 6*q^21 + 25*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3, {x, 0, n}]; (* Michael Somos, May 17 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3, n))}; /* Michael Somos, Jun 14 2007 */
(Magma) A := Basis( CuspForms( Gamma1(12), 3), 140); A[1] - 3*A[3]; /* Michael Somos, May 17 2015 */

Crossrefs

Cf. A007332, A152243, A209939.

Sequence in context: A099095 A061980 A059683 * A209939 A231101 A193084

Adjacent sequences: A030205 A030206 A030207 * A030209 A030210 A030211

Keyword

sign

Author

N. J. A. Sloane.

Status

approved