A181976 Expansion of a(q) * b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.

1, 0, -27, 72, 0, -216, 270, 0, -459, 720, 0, -1080, 936, 0, -1350, 2160, 0, -2592, 2214, 0, -2808, 3600, 0, -4752, 4590, 0, -4590, 6552, 0, -7560, 5184, 0, -7371, 10800, 0, -10800, 9360, 0, -9774, 12240, 0, -15120, 13500, 0, -14040, 17712, 0, -19872, 14760

Offset: 0

Author

Michael Somos, Apr 04 2012

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 27*q^2 + 72*q^3 - 216*q^5 + 270*q^6 - 459*q^8 + 720*q^9 + ...
		

Links

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

Crossrefs

Cf. A004007.

Programs

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^9 + 9*q*eta[q]^6*eta[q^9]^3)/eta[q^3]^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^9 + 9 * x * eta(x + A)^6 * eta(x^9 + A)^3) / eta(x^3 + A)^3, n))};

Formula

Expansion of b(q^3)^3 - 3 * b(q) * c(q^3)^2 in powers of q where b(), c() are cubic AGM theta functions.

Expansion of b(q^3)^2 * (b(q) + c(q^3)) in powers of q^3 where b(), c() are cubic AGM theta functions.

Expansion of (eta(q)^9 + 9 * q * eta(q)^6 * eta(q^9)^3) / eta(q^3)^3 in powers of q.

a(3*n + 1) = 0. a(3*n) = A004007(n).