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%I #56 Sep 08 2022 08:44:50
%S 1,-2,-3,4,6,6,-16,-8,9,-12,12,-12,38,32,-18,16,-126,-18,20,24,48,-24,
%T 168,24,-89,-76,-27,-64,30,36,-88,-32,-36,252,-96,36,254,-40,-114,-48,
%U 42,-96,-52,48,54,-336,-96,-48,-87,178,378,152,198,54,72,128
%N Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.
%C Identical to table 1, p. 493, of Alaca citation. - _Jonathan Vos Post_, May 24 2007
%C Unique cusp form of weight 4 for congruence group Gamma_1(6). - _Michael Somos_, Aug 11 2011
%C Number 14 of the 74 eta-quotients listed in Table I of Martin (1996).
%C The table 1, p. 493 of Alaca reference is the first 50 values of c_6(n). - _Michael Somos_, May 17 2015
%H Seiichi Manyama, <a href="/A030209/b030209.txt">Table of n, a(n) for n = 1..10000</a>
%H Saban Alaca and Kenneth Williams, <a href="http://dx.doi.org/10.1016/j.jnt.2006.10.004">Evaluation of the convolution sums sum_{l+6m=n} sigma(l)*sigma(m) and sum_{2l+3m=n} sigma(l)*sigma(m)</a>, J. Number Theory 124 (2007), no. 2, 491-510. MR2321376 (2008a:11005)
%H K. Bringmann and K. Ono, <a href="http://www.pnas.org/content/104/10/3725.abstract">Lifting cusp forms to Maass forms with an application to partitions</a>, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.
%H Brian Conrey and David Farmer (?), <a href="http://www.math.okstate.edu/~loriw/degree2/degree2hm/eta2/eta2.html">Eta Products and Quotients which are Newforms</a>.
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/6/4/0/">Newforms of weight 4 on Gamma_0(6)</a>.
%H 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 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F Euler transform of period 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - _Michael Somos_, Feb 13 2006
%F 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
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 11 2011
%F G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.
%F a(2*n) = -2 * a(n). Convolution square of A030188. - _Michael Somos_, May 27 2012
%F Convolution with A181102 is A186100. - _Michael Somos_, Jul 07 2015
%e G.f. = 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 + ...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* _Michael Somos_, Aug 11 2011 *)
%o (PARI) {a(n) = my(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 */
%o (Sage) CuspForms( Gamma1(6), 4, prec = 57).0; # _Michael Somos_, Aug 11 2011
%o (Magma) Basis( CuspForms( Gamma1(6), 4), 57) [1]; /* _Michael Somos_, May 17 2015 */
%Y Cf. A030188, A181101, A186100.
%K sign,mult
%O 1,2
%A _N. J. A. Sloane_.
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