A321466 Expansion of (phi(x^3)^3 / phi(x))^2 in powers of x where phi() is a Ramanujan theta function.

1, -4, 12, -20, 28, -24, 28, -32, 60, -68, 72, -48, 44, -56, 96, -120, 124, -72, 76, -80, 168, -160, 144, -96, 76, -124, 168, -212, 224, -120, 168, -128, 252, -240, 216, -192, 92, -152, 240, -280, 360, -168, 224, -176, 336, -408, 288, -192, 140, -228, 372

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

The g.f. of A113973 is k = k(q) := phi(x^3)^3 / phi(x) given in equation (2.2) page 996 of Williams 2012, and the g.f. of k^2 which is given in equation (2.3) page 997 is this sequence.

Number 54 of the 126 eta-quotients listed in Table 1 of Williams 2012.

Links

Table of n, a(n) for n=0..50.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Formula

Expansion of (eta(q)^2 * eta(q^4)^2 * eta(q^6)^15 / (eta(q^2)^5 * eta(q^3)^6 * eta(q^12)^6))^2 in powers of q.

Expansion of ((a(x) - 2*a(x^2) - 2*a(x^4))/3)^2 = ((b(x) + 2*b(x^4))^2 / (9*b(x^2)))^2 in powers of x where a(), b() are cubic AGM theta functions.

Euler transform of period 12 sequence [-4, 6, 8, 2, -4, -12, -4, 2, 8, 6, -4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321465.

G.f.: (theta_3(0, x^3)^3 / theta_3(0, x))^2 where theta_3(0, x) is a Jacobi theta function.

G.f.: (Product_{k>0} f(x^k))^2 where f(x) := ((1 - x) * (1 + x^2))^2 * ((1 - x^3) * (1 + x^3)^3)^3 / ((1 - x^2) * (1 + x^6)^2)^3.

a(n) = -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0.

a(n) = (-1)^n * A227226(n). Convolution square of A113973.

Example

G.f. = 1 - 4*x + 12*x^2 - 20*x^3 + 28*x^4 - 24*x^5 + 28*x^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3]^6 / EllipticTheta[ 3, 0, x]^2, {x, 0, n}];
a[ n_] := With[{s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n == 0], -4 (s[n] - 6 s[n/2] + s[n/3] + 4 s[n/4] + 2 s[n/6] + 4 s[n/12])]];

Prog

(PARI) {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, -4*(s(n) - 6*s(n/2) + s(n/3) + 4*s(n/4) + 2*s(n/6) + 4*s(n/12)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^15 / (eta(x^2 + A)^5 * eta(x^3 + A)^6 * eta(x^12 + A)^6))^2, n))};
(Magma) A := Basis( ModularForms( Gamma0(12), 2), 51); A[1] - 4*A[2] + 12*A[3] - 20*A[4] + 28*A[5];

Crossrefs

Cf. A113973,A227226,A321465.

Sequence in context: A100717 A365883 A285526 * A227226 A242118 A030387

Adjacent sequences: A321463 A321464 A321465 * A321467 A321468 A321469

Keyword

sign

Author

Michael Somos, Nov 11 2018

Status

approved