A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 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, 360, 392, 216, 220, 288

Offset: 0

Author

Michael Somos, Sep 19 2013

Keywords

• nonn

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 13 and 34 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

Examples

			G.f. = 1 + 4*q + 12*q^2 + 20*q^3 + 28*q^4 + 24*q^5 + 28*q^6 + 32*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• 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.

Crossrefs

Cf. A123330, A227229.

Programs

Magma

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 4*A[2] + 12*A[3];

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3]^6 / (QPochhammer[ q]^2 QPochhammer[ q^6]^3))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^6 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 4/3, 1, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 2, 1, 1, -6}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3))^2, n))};

Sage

A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 4*A[1] + 12*A[2];

Formula

Expansion of (a(q) + 2*a(q^2))^2 / 9 in powers of q where a(q) is a cubic AGM theta function.

Expansion of c(q)^4 / (3 * c(q^2))^2 in powers of q where c(q) is a cubic AGM theta function.

Expansion of (eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3))^2 in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227229.

Convolution square of A123330.