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A007332
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Expansion of 6-dimensional cusp form (eta(z)*eta(3z))^6.
(Formerly M4075)
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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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.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
W. Stein, Modular Forms Database.
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FORMULA
| G.f.: q Product (1-q^k)^6 (1-q^3k)^6.
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
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EXAMPLE
| q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 - 54*q^6 - 40*q^7 + ...
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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PROG
| (PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((eta(x+A)*eta(x^3+A))^6, n)) /* Michael Somos, Jul 16 2004 */
(PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff(( prod(k=1, n, (1-(k%3==0)*x^k)*(1-x^k), 1+A) )^6, n)) /* Michael Somos, Jul 16 2004 */
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6, n))} /* Michael Somos Nov 16 2008 */
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CROSSREFS
| Convolution square of A030208.
Sequence in context: A201130 A073240 A019853 * A131691 A021063 A110649
Adjacent sequences: A007329 A007330 A007331 * A007333 A007334 A007335
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KEYWORD
| sign,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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