%I #7 Sep 08 2022 08:46:24

%S 0,1,-9,31,-45,6,45,8,-117,121,-54,12,9,14,-72,186,-261,18,207,20,

%T -270,248,-108,24,-63,31,-126,391,-360,30,270,32,-549,372,-162,48,171,

%U 38,-180,434,-702,42,360,44,-540,726,-216,48,-207,57,-279,558,-630,54,693

%N Expansion of x * (psi(x^6) / psi(-x^3))^3 * phi(-x)^5 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

%C Number 105 of the 126 eta-quotients listed in Table 1 of Williams 2012.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A328788.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H 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 Euler transform of period 12 sequence [-9, -5, -6, -4, -9, -2, -9, -4, -6, -5, -9, -4, ...].

%F Expansion of x * phi(-x)^5 / psi(-x) * (f(-x^12) / f(-x^3))^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%F Expansion of eta(q)^9 * eta(q^12)^3 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^4)) in powers of q.

%F a(n) = s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4) if n>0 where s(x) = sum of divisors of x for integer x else 0.

%F a(n) = -(-1)^n * A133739(n). a(4*n + 2) = -9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

%e G.f. = x - 9*x^2 + 31*x^3 - 45*x^4 + 6*x^5 + 45*x^6 + 8*x^7 - 117*x^8 + ...

%t a[ n_] := SeriesCoefficient[ 1/2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)])^3 EllipticTheta[ 4, 0, x]^5 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}] // PowerExpand;

%o (PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, 0, s(n) - 12*s(n/2) + 27*s(n/3) - 16*s(n/4))};

%o (PARI) {a(n) = my(A); if( n < 1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^9 * eta(x^12 + A)^3 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^4 + A)), n))};

%o (Magma) A := Basis( ModularForms( Gamma0(12), 2), 52); A[2] - 9*A[3] + 31*A[4] - 45*A[5];

%Y Cf. A000203, A098098, A133739, A134077, A328788.

%K sign

%O 0,3

%A _Michael Somos_, Nov 18 2019