A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

1, -2, -3, 4, 6, 6, -16, -8, 9, -12, 12, -12, 38, 32, -18, 16, -126, -18, 20, 24, 48, -24, 168, 24, -89, -76, -27, -64, 30, 36, -88, -32, -36, 252, -96, 36, 254, -40, -114, -48, 42, -96, -52, 48, 54, -336, -96, -48, -87, 178, 378, 152, 198, 54, 72, 128

Offset

1,2

Comments

Identical to table 1, p. 493, of Alaca citation. - Jonathan Vos Post, May 24 2007

Unique cusp form of weight 4 for congruence group Gamma_1(6). - Michael Somos, Aug 11 2011

Number 14 of the 74 eta-quotients listed in Table I of Martin (1996).

The table 1, p. 493 of Alaca reference is the first 50 values of c_6(n). - Michael Somos, May 17 2015

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Saban Alaca and Kenneth Williams, Evaluation of the convolution sums sum_{l+6m=n} sigma(l)*sigma(m) and sum_{2l+3m=n} sigma(l)*sigma(m), J. Number Theory 124 (2007), no. 2, 491-510. MR2321376 (2008a:11005)

K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.

Brian Conrey and David Farmer (?), Eta Products and Quotients which are Newforms.

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

LMFDB, Newforms of weight 4 on Gamma_0(6).

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 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - Michael Somos, Feb 13 2006

a(n) is multiplicative with a(p^e) = (-p)^e if p<5, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011

G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.

a(2*n) = -2 * a(n). Convolution square of A030188. - Michael Somos, May 27 2012

Convolution with A181102 is A186100. - Michael Somos, Jul 07 2015

Example

G.f. = q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + 6*q^6 - 16*q^7 - 8*q^8 + 9*q^9 - 12*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A))^2, n))}; /* Michael Somos, Feb 14 2006 */
(Sage) CuspForms( Gamma1(6), 4, prec = 57).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(6), 4), 57) [1]; /* Michael Somos, May 17 2015 */

Crossrefs

Cf. A030188, A181101, A186100.

Sequence in context: A135510 A265398 A299438 * A265399 A138588 A344412

Adjacent sequences: A030206 A030207 A030208 * A030210 A030211 A030212

Keyword

sign,mult

Author

N. J. A. Sloane.

Status

approved