cp's OEIS Frontend

A007332 Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.

%I A007332 M4075 #44 Sep 08 2022 08:44:35
%S A007332 0,1,-6,9,4,6,-54,-40,168,81,-36,-564,36,638,240,54,-1136,882,-486,
%T A007332 -556,24,-360,3384,-840,1512,-3089,-3828,729,-160,4638,-324,4400,1440,
%U A007332 -5076,-5292,-240,324,-2410,3336,5742,1008,-6870,2160,9644,-2256,486,5040
%N A007332 Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.
%C A007332 Number 5 of the 74 eta-quotients listed in Table I of Martin (1996).
%D A007332 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 204.
%D A007332 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 145, problem 13.
%D A007332 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007332 Seiichi Manyama, <a href="/A007332/b007332.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H A007332 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A007332 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A007332 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A007332 G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6.
%F A007332 Expansion of (eta(q) * eta(q^3))^6 in powers of q. - _Michael Somos_, Jul 16 2004
%F A007332 Euler transform of period 3 sequence [ -6, -6, -12, ...]. - _Michael Somos_, Jul 16 2004
%F A007332 Expansion of a newform of level 3, weight 6 and trivial character. - _Michael Somos_, Nov 16 2008
%F A007332 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
%F A007332 Given A = A0 + A1 + A2 is the 3-section, then 0 = A2^2 - 4 * A1*A0. - _Michael Somos_, Mar 08 2006
%F A007332 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
%F A007332 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
%F A007332 a(3*n) = 9 * a(n). - _Michael Somos_, Nov 16 2008
%F A007332 Convolution square of A030208.
%e A007332 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 + ...
%t A007332 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3] )^6, {q, 0, n}]; (* _Michael Somos_, May 28 2013 *)
%o A007332 (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 */
%o A007332 (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 */
%o A007332 (Sage) CuspForms( Gamma0(3), 6, prec=47).0; # _Michael Somos_, May 28 2013
%o A007332 (Magma) Basis( CuspForms( Gamma0(3), 6), 47) [1]; /* _Michael Somos_, Dec 10 2013 */
%Y A007332 Cf. A030208.
%K A007332 sign,easy,nice,mult
%O A007332 0,3
%A A007332 _N. J. A. Sloane_, _Mira Bernstein_