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A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 where phi() is a Ramanujan theta function.

%I A227226 #23 Feb 16 2025 08:33:20
%S A227226 1,4,12,20,28,24,28,32,60,68,72,48,44,56,96,120,124,72,76,80,168,160,
%T A227226 144,96,76,124,168,212,224,120,168,128,252,240,216,192,92,152,240,280,
%U A227226 360,168,224,176,336,408,288,192,140,228,372,360,392,216,220,288
%N A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 where phi() is a Ramanujan theta function.
%C A227226 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A227226 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A227226 Number 13 and 34 of the 126 eta-quotients listed in Table 1 of Williams 2012. - _Michael Somos_, Nov 10 2018
%H A227226 G. C. Greubel, <a href="/A227226/b227226.txt">Table of n, a(n) for n = 0..2500</a>
%H A227226 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227226 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A227226 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 A227226 Expansion of (a(q) + 2*a(q^2))^2 / 9 in powers of q where a(q) is a cubic AGM theta function.
%F A227226 Expansion of c(q)^4 / (3 * c(q^2))^2 in powers of q where c(q) is a cubic AGM theta function.
%F A227226 Expansion of (eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3))^2 in powers of q.
%F A227226 Euler transform of period 6 sequence [4, 2, -8, 2, 4, -4, ...].
%F A227226 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227229.
%F A227226 Convolution square of A123330.
%e A227226 G.f. = 1 + 4*q + 12*q^2 + 20*q^3 + 28*q^4 + 24*q^5 + 28*q^6 + 32*q^7 + ...
%t A227226 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3]^6 / (QPochhammer[ q]^2 QPochhammer[ q^6]^3))^2, {q, 0, n}];
%t A227226 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^6 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
%t A227226 a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 4/3, 1, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
%t A227226 a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 2, 1, 1, -6}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
%o A227226 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3))^2, n))};
%o A227226 (Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 4*A[1] + 12*A[2];
%o A227226 (Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 4*A[2] + 12*A[3];
%Y A227226 Cf. A123330, A227229.
%K A227226 nonn
%O A227226 0,2
%A A227226 _Michael Somos_, Sep 19 2013