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

1, 12, 60, 156, 204, 72, -84, 96, 492, 588, 360, 144, 60, 168, 480, 936, 1068, 216, -516, 240, 1224, 1248, 720, 288, 348, 372, 840, 1884, 1632, 360, -504, 384, 2220, 1872, 1080, 576, -372, 456, 1200, 2184, 2952, 504, -672, 528, 2448, 3528, 1440, 576, 924, 684

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).

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

Links

Table of n, a(n) for n=0..49.

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)^30 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^12 * eta(q^4)^12 * eta(q^6)^10) in powers of q.

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.

Euler transform of period 12 sequence [12, -18, 8, -6, 12, -12, 12, -6, 8, -18, 12, -4, ...].

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.

G.f.: (theta_3(0, x)^3 / theta_3(0, x^3))^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)^6 * (1 - x^2)^3 * (1 + x^6)^2) / ((1 + x^2)^6 * (1 - x^3) * (1 + x^3)^3).

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.

a(n) = (-1)^n * A229616(n). Convolution square of A113660.

Example

G.f. = 1 + 12*x + 60*x^2 + 156*x^3 + 204*x^4 + 72*x^5 - 84*x^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 / EllipticTheta[ 3, 0, x^3]^2, {x, 0, n}];
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])]];

Prog

(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)))};
(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))};
(Magma) A := Basis( ModularForms( Gamma0(12), 2), 50); A[1] + 12*A[2] + 60*A[3] + 156*A[4] + 204*A[5];

Crossrefs

Cf. A113660, A229616, A321466.

Sequence in context: A158443 A153792 A229616 * A000141 A328094 A300758

Adjacent sequences: A321462 A321463 A321464 * A321466 A321467 A321468

Keyword

sign

Author

Michael Somos, Nov 11 2018

Status

approved