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A030209
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Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.
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0
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1, -2, -3, 4, 6, 6, -16, -8, 9, -12, 12, -12, 38, 32, -18, 16, -126, -18, 20, 24, 48, -24, 168, 24, -89, -76, -27, -64, 30, 36, -88, -32, -36, 252, -96, 36, 254, -40, -114, -48, 42, -96, -52, 48, 54, -336, -96, -48, -87, 178, 378, 152, 198, 54, 72, 128, -60, -60, -660, -72, -538, 176, -144, 64, 228, 72, 884
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Identical to table 1, p. 493, of Alaca citation. - Jonathan Vos Post (jvospost3(AT)gmail.com), May 24 2007
Unique cusp form of weight 4 for congruence group Gamma_1(6). - Michael Somos, Aug 11 2011
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REFERENCES
| Saban Alaca and Kenneth Williams, Evaluation of the convolution sums..., Journal of Number Theory 124(2007)491-510.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.
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LINKS
| Author?, Eta Products and Quotients which are Newforms.
K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions.
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FORMULA
| Euler transform of period 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - Michael Somos, Feb 13 2006
a(n) is multiplicative with a(p^e) = (-p)^e if p<5, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 g(t) where q = exp(2 pi i t). - Michael Somos, Aug 11 2011
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.
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EXAMPLE
| q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + 6*q^6 - 16*q^7 - 8*q^8 + 9*q^9 - 12*q^10 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q] QPochhammer[ q^2, q^2] QPochhammer[ q^3, q^3] QPochhammer[ q^6, q^6])^2, {q, 0, n}] (* Michael Somos, Aug 11 2011 *)
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x+ A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A))^2, n))} /* Michael Somos, Feb 14 2006 */
(SAGE) CuspForms( Gamma1(6), 4, prec = 100). 0 # Michael Somos, Aug 11 2011
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CROSSREFS
| Sequence in context: A000793 A062163 A002729 * A138588 A143102 A013944
Adjacent sequences: A030206 A030207 A030208 * A030210 A030211 A030212
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| added ref to Bringmann and Ono
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