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%I #20 Sep 08 2022 08:45:28
%S 1,-3,-2,12,-4,-9,6,-12,8,3,4,12,-16,33,-24,-60,7,27,8,24,44,-24,18,
%T -36,-34,-48,-12,84,-40,9,24,48,-33,21,-16,0,72,-111,-6,36,50,-96,-8,
%U -120,8,111,-24,96,-16,195,32,-132,-76,48,-66,24,-54,-123,48,-144,-32,102,120,-24,176,117,-14,-12,-28,-192,-54
%N Expansion of q^(-2/3) * b(q) * b(q^2) * c(q^2) / 3 in powers of q where b(), c() are cubic AGM theta functions.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H Seiichi Manyama, <a href="/A122407/b122407.txt">Table of n, a(n) for n = 0..1000</a>
%H N. Elkies, <a href="http://mathoverflow.net/questions/84443/">Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418</a>, MathOverflow.
%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/18/3/17/a/">Newform 18.3.17.a</a>.
%F Expansion of q^(-2/3) * eta(q)^3 * eta(q^2)^2 * eta(q^6)^2 / eta(q^3) in powers of q.
%F Euler transform of period 6 sequence [-3, -5, -2, -5, -3, -6, ...].
%F Expansion of a newform level 18 weight 3 and character chi_18(17,.). - _Michael Somos_, Jan 08 2017
%F a(n) = sqrt(-2) * b(3*n + 2) where b() is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - Kronecker(-12, p) * p^2 * b(p^(e-2)). - _Michael Somos_, Jan 08 2017
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 972^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208385.
%F G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^4 * (1 - x^k + x^(2*k))^2 *(1 + x^k + x^(2*k)).
%F a(n) = A208384(3*n + 2). -2 * a(n) = A208385(3*n + 2).
%F -2 * a(n) = A116418(2*n + 1) = a(4*n + 2). a(2*n) = A116418(n). - _Michael Somos_, Jan 08 2017
%e G.f. = 1 - 3*x - 2*x^2 + 12*x^3 - 4*x^4 - 9*x^5 + 6*x^6 - 12*x^7 + 8*x^8 + ...
%e G.f. = q^2 - 3*q^5 - 2*q^8 + 12*q^11 - 4*q^14 - 9*q^17 + 6*q^20 - 12*q^23 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^2]^2 QPochhammer[ x^6]^2 / QPochhammer[ x^3], {x, 0, n}]; (* _Michael Somos_, Jan 08 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^2 + A)^2 * eta(x^6 + A)^2 / eta(x^3 + A), n))};
%o (Magma) A := Basis( CuspForms( Gamma1(18), 3), 211); A[2] - 3*A[5] - 2*A[8]; /* _Michael Somos_, Jan 08 2017 */
%Y Cf. A116418, A208384, A208385.
%K sign
%O 0,2
%A _Michael Somos_, Sep 02 2006
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