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A007332
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Expansion of 6-dimensional cusp form (eta(q) * eta(q^3))^6 in powers of q.
(Formerly M4075)
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10
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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
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OFFSET
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0,3
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COMMENTS
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Number 5 of the 74 eta-quotients listed in Table I of Martin (1996).
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3] )^6, {q, 0, n}]; (* Michael Somos, May 28 2013 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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sign,easy,nice,mult
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AUTHOR
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STATUS
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approved
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