A281722 Expansion of r(q) * s(q) in powers of q where r(), s() are cubic AGM functions.

1, 3, -18, 12, 21, -36, 36, 24, -90, 12, 54, -72, 84, 42, -144, 72, 93, -108, 36, 60, -252, 96, 108, -144, 180, 93, -252, 12, 168, -180, 216, 96, -378, 144, 162, -288, 84, 114, -360, 168, 270, -252, 288, 132, -504, 72, 216, -288, 372, 171, -558, 216, 294, -324

Offset: 0

Author

Michael Somos, Jan 28 2017

Keywords

• sign

Comments

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

Examples

			G.f. = 1 + 3*q - 18*q^2 + 12*q^3 + 21*q^4 - 36*q^5 + 36*q^6 + 24*q^7 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from G. C. Greubel)

Crossrefs

Cf. A004016, A005928, A033686, A096726, A144614.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 (QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3]^2, {q, 0, n}];

PARI

{a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A)^3 * (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A)^2, n))};

Formula

Convolution of the sequences A004016 and A005928.

The g.f. is the product of the g.f.'s for A004016 and A005928. - N. J. A. Sloane, Jan 30 2017

Expansion of eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144614.

a(3*n + 2) = A096726(3*n + 2) - 27 * A033686(n). a(n) == A096726(n) (mod 27). - Michael Somos, Sep 04 2017