A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

1, 6, 15, 24, 33, 36, 33, 48, 69, 78, 90, 72, 51, 84, 120, 144, 141, 108, 87, 120, 198, 192, 180, 144, 87, 186, 210, 240, 264, 180, 198, 192, 285, 288, 270, 288, 105, 228, 300, 336, 414, 252, 264, 264, 396, 468, 360, 288, 159, 342, 465, 432, 462, 324, 249

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 8 and 32 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

Examples

			G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...
		

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. A107760, A227226.

Programs

Magma

A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ 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)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2))^2, n))};

Sage

A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];

Formula

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.

Convolution square of A107760.

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