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%I #28 Sep 08 2022 08:46:08
%S 1,-6,9,12,-42,18,36,-48,45,12,-108,36,84,-84,72,72,-186,54,36,-120,
%T 126,96,-216,72,180,-186,126,12,-336,90,216,-192,189,144,-324,144,84,
%U -228,180,168,-540,126,288,-264,252,72,-432,144,372,-342,279,216,-588
%N Expansion of b(q)^2 in powers of q where b() is a cubic AGM theta function.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See t, page 1.
%H Seiichi Manyama, <a href="/A242874/b242874.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2500 from G. C. Greubel)
%F Expansion of (eta(q)^3 / eta(q^3))^2 in powers of q.
%F Euler transform of period 3 sequence [-6, -6, -4, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 243 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033686.
%F G.f.: Product_{k>0} ( (1 - x^k)^3 / (1 - x^(3*k)) )^2.
%F a(3*n) = A008653(n). a(3*n + 1) = -6 * A144614(n). a(3*n + 2) = 9 * A033686(n).
%F Convolution square of A005928.
%e G.f. = 1 - 6*q + 9*q^2 + 12*q^3 - 42*q^4 + 18*q^5 + 36*q^6 - 48*q^7 + 45*q^8 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^2, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^2, n))};
%o (Sage) A = ModularForms( Gamma0(9), 2, prec=53) . basis(); A[0] - 6*A[1] + 9*A[2];
%o (Magma) A := Basis( ModularForms( Gamma0(9), 2), 53); A[1] - 6*A[2] + 9*A[3]; /* _Michael Somos_, Sep 27 2016 */
%Y Cf. A005928, A008653, A033686, A144614.
%K sign
%O 0,2
%A _Michael Somos_, May 26 2014
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