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A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.

%I A321528 #7 Feb 16 2025 08:33:57
%S A321528 1,-6,6,30,-66,-36,186,-48,-210,138,36,-72,114,-84,48,180,-498,-108,
%T A321528 726,-120,-396,240,72,-144,-30,-186,84,462,-528,-180,1116,-192,-1074,
%U A321528 360,108,-288,654,-228,120,420,-1260,-252,1488,-264,-792,828,144,-288,-318
%N A321528 Expansion of b(x)^2 * b(x^2) / b(x^4) where b is a cubic AGM theta function.
%C A321528 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A321528 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A321528 Number 64 of the 126 eta-quotients listed in Table 1 of Williams 2012.
%H A321528 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A321528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A321528 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 A321528 Expansion of phi(-x) * phi(-x^2)^3 / (phi(-x^3) * phi(-x^6)) in powers of x where phi() is a Ramanujan theta function.
%F A321528 Expansion of eta(q)^6 * eta(q^2)^3 * eta(q^12) / (eta(q^3)^2 * eta(q^4)^3 * eta(q^6)) in powers of q.
%F A321528 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 864 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321527.
%F A321528 a(n) = -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)) if n>0, where s(x) = sum of divisors of x for integer x else 0.
%F A321528 a(2*n + 1) = -6 * A134077(n). a(6*n + 5) = -a(12*n + 10) = -36 * A098098(n).
%e A321528 G.f. = 1 - 6*x + 6*x^2 + 30*x^3 - 66*x^4 - 36*x^5 + 186*x^6 - 48*x^7 + ...
%t A321528 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^2])^3 / ( EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];
%t A321528 a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n==0], -6 (s[n/1] - 4 s[n/2] - 9 s[n/3] + 16 s[n/4])]];
%t A321528 a[ n_] := If[ n < 1, Boole[n==0], -6 Sum[ d {1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1, 0}[[Mod[d, 12, 1]]], {d, Divisors[n]}]];
%o A321528 (PARI) {a(n) = if( n<1, n==0, -6 * sumdiv( n, d, d * [0, 1, -1, -2, 3, 1, -4, 1, 3, -2, -1, 1][d%12 + 1]))};
%o A321528 (PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -6 * (s(n/1) - 4*s(n/2) - 9*s(n/3) + 16*s(n/4)))};
%o A321528 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^2 + A)^3 * eta(x^12 + A) / (eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
%o A321528 (Magma) A := Basis( ModularForms( Gamma0(12), 2), 49); A[1] - 6*A[2] + 6*A[3] + 30*A[4] - 66*A[5];
%Y A321528 Cf. A098098, A134077, A321527.
%K A321528 sign
%O A321528 0,2
%A A321528 _Michael Somos_, Nov 12 2018