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A007332 Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.
(Formerly M4075)
4
0, 1, -6, 9, 4, 6, -54, -40, 168, 81, -36, -564, 36, 638, 240, 54, -1136, 882, -486, -556, 24, -360, 3384, -840, 1512, -3089, -3828, 729, -160, 4638, -324, 4400, 1440, -5076, -5292, -240, 324, -2410, 3336, 5742, 1008, -6870, 2160, 9644, -2256, 486, 5040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number 5 of the 74 eta-quotients listed in Table I of Martin (1996).

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 204.

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 145, problem 13.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

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

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

Expansion of (eta(q) * eta(q^3))^6 in powers of q. - Michael Somos, Jul 16 2004

Euler transform of period 3 sequence [ -6, -6, -12, ...]. - Michael Somos, Jul 16 2004

Expansion of a newform of level 3, weight 6 and trivial character. - Michael Somos, Nov 16 2008

a(n) is multiplicative with a(3^e) = 9^e, a(p^e) = a(p) * a(p^(e-1)) - p^5 * a(p^(e-2)). - Michael Somos, Mar 08 2006

Given A = A0 + A1 + A2 is the 3-section, then 0 = A2^2 - 4 * A1*A0. - Michael Somos, Mar 08 2006

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u * w * (u + 12 * v + 64 * w) - v^3. - Michael Somos, May 02 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^3 (t/i)^6 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008

a(3*n) = 9 * a(n). - Michael Somos, Nov 16 2008

Convolution square of A030208.

EXAMPLE

G.f. = q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 - 54*q^6 - 40*q^7 + 168*q^8 + 81*q^9 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3] )^6, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6, n))}; /* Michael Somos, Jul 16 2004 */

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( prod( k=1, n, (1 - (k%3==0) * x^k) * (1 - x^k), 1 + A) )^6, n))}; /* Michael Somos, Jul 16 2004 */

(Sage) CuspForms( Gamma0(3), 6, prec=47).0; # Michael Somos, May 28 2013

(MAGMA) Basis( CuspForms( Gamma0(3), 6), 47) [1]; /* Michael Somos, Dec 10 2013 */

CROSSREFS

Cf. A030208.

Sequence in context: A248759 A073240 A019853 * A246041 A131691 A258504

Adjacent sequences:  A007329 A007330 A007331 * A007333 A007334 A007335

KEYWORD

sign,easy,nice,mult

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)