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A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

%I A030208 #35 Jul 08 2025 19:31:00
%S A030208 1,-3,0,2,9,0,-22,0,0,26,-6,0,25,-27,0,-46,0,0,26,66,0,-22,0,0,-45,0,
%T A030208 0,0,-78,0,74,18,0,122,0,0,-46,-75,0,-142,81,0,0,0,0,-44,138,0,2,0,0,
%U A030208 194,0,0,-214,-78,0,0,-198,0,121,0,0,146,66,0,52,0,0,-22
%N A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.
%C A030208 Number 28 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A030208 Expansion of a newform level 12 weight 3 and character [0, 1].
%H A030208 Seiichi Manyama, <a href="/A030208/b030208.txt">Table of n, a(n) for n = 0..10000</a>
%H A030208 Author?, <a href="http://www.math.okstate.edu/~loriw/degree2/degree2hm/eta2/eta2.html">Eta Products and Quotients which are Newforms</a>. [Broken link?]
%H A030208 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A030208 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 A030208 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A030208 Euler transform of period 3 sequence [ -3, -3, -6, ...]. - _Michael Somos_, Feb 13 2006
%F A030208 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-3)^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise. - _Michael Somos_, Feb 13 2006
%F A030208 G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^3.
%F A030208 G.f.: Sum_{k>=0} a(k) * q^(2*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * q^(u*u + 3*v*v). - _Michael Somos_, Jun 14 2007
%F A030208 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 16 2008
%F A030208 a(3*n + 1) = -3 * a(n). a(3*n + 2) = 0. a(3*n) = A152243(n). - _Michael Somos_, Mar 09 2012
%F A030208 a(n) = (-1)^n * A209939(n). - _Michael Somos_, Mar 16 2012
%F A030208 Convolution square is A007332. - _Michael Somos_, Nov 16 2008
%e A030208 G.f. = 1 - 3*x + 2*x^3 + 9*x^4 - 22*x^6 + 26*x^9 - 6*x^10 + 25*x^12 - 27*x^13 + ...
%e A030208 G.f. = q - 3*q^3 + 2*q^7 + 9*q^9 - 22*q^13 + 26*q^19 - 6*q^21 + 25*q^25 + ...
%t A030208 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3, {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%o A030208 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3, n))}; /* _Michael Somos_, Jun 14 2007 */
%o A030208 (Magma) A := Basis( CuspForms( Gamma1(12), 3), 140); A[1] - 3*A[3]; /* _Michael Somos_, May 17 2015 */
%Y A030208 Cf. A007332, A152243, A209939.
%K A030208 sign
%O A030208 0,2
%A A030208 _N. J. A. Sloane_