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A321465 Expansion of (phi(x)^3 / phi(x^3))^2 in powers of x where phi() is a Ramanujan theta function.

%I A321465 #10 Feb 16 2025 08:33:57
%S A321465 1,12,60,156,204,72,-84,96,492,588,360,144,60,168,480,936,1068,216,
%T A321465 -516,240,1224,1248,720,288,348,372,840,1884,1632,360,-504,384,2220,
%U A321465 1872,1080,576,-372,456,1200,2184,2952,504,-672,528,2448,3528,1440,576,924,684
%N A321465 Expansion of (phi(x)^3 / phi(x^3))^2 in powers of x where phi() is a Ramanujan theta function.
%C A321465 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A321465 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A321465 Number 1 of the 126 eta-quotients listed in Table 1 of Williams 2012.
%H A321465 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A321465 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A321465 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 A321465 Expansion of eta(q^2)^30 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^12 * eta(q^4)^12 * eta(q^6)^10) in powers of q.
%F A321465 Expansion of ((a(x) + 2*a(x^2) - 2*a(x^4))/3)^2 = (b(-x)^2 / b(x^2))^2 in powers of x where a(), b() are cubic AGM theta functions.
%F A321465 Euler transform of period 12 sequence [12, -18, 8, -6, 12, -12, 12, -6, 8, -18, 12, -4, ...].
%F A321465 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A342166.
%F A321465 G.f.: (theta_3(0, x)^3 / theta_3(0, x^3))^2 where theta_3(0, x) is a Jacobi theta function.
%F A321465 G.f.: (Product_{k>0} f(x^k))^2 where f(x) := ((1 + x)^6 * (1 - x^2)^3 * (1 + x^6)^2) / ((1 + x^2)^6 * (1 - x^3) * (1 + x^3)^3).
%F A321465 a(n) = 12*(s(n) + 2*s(n/2) + 9*s(n/3) + 4*s(n/4) - 54*s(n/6) + 36*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0.
%F A321465 a(n) = (-1)^n * A229616(n). Convolution square of A113660.
%e A321465 G.f. = 1 + 12*x + 60*x^2 + 156*x^3 + 204*x^4 + 72*x^5 - 84*x^6 + ...
%t A321465 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 / EllipticTheta[ 3, 0, x^3]^2, {x, 0, n}];
%t A321465 a[ n_] := With[{s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n == 0], 12 (s[n] + 2 s[n/2] + 9 s[n/3] + 4 s[n/4] - 54 s[n/6] + 36 s[n/12])]];
%o A321465 (PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, 12*(s(n) + 2*s(n/2) + 9*s(n/3) + 4*s(n/4) - 54*s(n/6) + 36*s(n/12)))};
%o A321465 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^15 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^6 * eta(x^6 + A)^5))^2, n))};
%o A321465 (Magma) A := Basis( ModularForms( Gamma0(12), 2), 50); A[1] + 12*A[2] + 60*A[3] + 156*A[4] + 204*A[5];
%Y A321465 Cf. A113660, A229616, A321466.
%K A321465 sign
%O A321465 0,2
%A A321465 _Michael Somos_, Nov 11 2018