A242874 Expansion of b(q)^2 in powers of q where b() is a cubic AGM theta function.

1, -6, 9, 12, -42, 18, 36, -48, 45, 12, -108, 36, 84, -84, 72, 72, -186, 54, 36, -120, 126, 96, -216, 72, 180, -186, 126, 12, -336, 90, 216, -192, 189, 144, -324, 144, 84, -228, 180, 168, -540, 126, 288, -264, 252, 72, -432, 144, 372, -342, 279, 216, -588

Offset: 0

Author

Michael Somos, May 26 2014

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 6*q + 9*q^2 + 12*q^3 - 42*q^4 + 18*q^5 + 36*q^6 - 48*q^7 + 45*q^8 + ...
		

References

• O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See t, page 1.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

Crossrefs

Cf. A005928, A008653, A033686, A144614.

Programs

Magma

A := Basis( ModularForms( Gamma0(9), 2), 53); A[1] - 6*A[2] + 9*A[3]; /* Michael Somos, Sep 27 2016 */

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^2, {q, 0, n}];

PARI

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

Sage

A = ModularForms( Gamma0(9), 2, prec=53) . basis(); A[0] - 6*A[1] + 9*A[2];

Formula

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

Euler transform of period 3 sequence [-6, -6, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 243 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033686.

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

a(3*n) = A008653(n). a(3*n + 1) = -6 * A144614(n). a(3*n + 2) = 9 * A033686(n).

Convolution square of A005928.