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A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

%I A227229 #21 Feb 16 2025 08:33:20
%S A227229 1,6,15,24,33,36,33,48,69,78,90,72,51,84,120,144,141,108,87,120,198,
%T A227229 192,180,144,87,186,210,240,264,180,198,192,285,288,270,288,105,228,
%U A227229 300,336,414,252,264,264,396,468,360,288,159,342,465,432,462,324,249
%N A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.
%C A227229 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A227229 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A227229 Number 8 and 32 of the 126 eta-quotients listed in Table 1 of Williams 2012. - _Michael Somos_, Nov 10 2018
%H A227229 G. C. Greubel, <a href="/A227229/b227229.txt">Table of n, a(n) for n = 0..2500</a>
%H A227229 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227229 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A227229 K. S. Williams, <a href="http://dx.doi.org/10.1142/S1793042112500595">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
%F A227229 Expansion of (a(q) + a(q^2))^2 / 4 in powers of q where a() is a cubic AGM theta function.
%F A227229 Expansion of (b(q^2)^2 / b(q))^2 in powers of q where b() is a cubic AGM theta function.
%F A227229 Expansion of (eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2))^2 in powers of q.
%F A227229 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.
%F A227229 Convolution square of A107760.
%F A227229 Euler transform of period 6 sequence [6, -6, 4, -6, 6, -4, ...].
%e A227229 G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...
%t A227229 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
%t A227229 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
%t A227229 a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
%t A227229 a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
%o A227229 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2))^2, n))};
%o A227229 (Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];
%o A227229 (Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];
%Y A227229 Cf. A107760, A227226.
%K A227229 nonn
%O A227229 0,2
%A A227229 _Michael Somos_, Sep 19 2013