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.
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
KEYWORD
sign,easy,nice,mult
AUTHOR
STATUS
approved