A279005 Expansion of 3 * q * b(q^9)^3 / c(q^3) in powers of q^3 where b(), c() are cubic AGM theta functions.

1, -1, -1, -6, 6, 7, 9, -8, -15, 13, -19, -4, -49, 57, 61, 32, -14, -75, 47, -98, -23, -124, 130, 152, 116, -37, -182, 96, -168, 0, -335, 352, 342, 196, -117, -390, 230, -440, -107, -600, 637, 671, 480, -184, -704, 469, -727, -112, -1235, 1241, 1291, 722, -341

Offset: 0

Author

Michael Somos, Dec 10 2016

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x - x^2 - 6*x^3 + 6*x^4 + 7*x^5 + 9*x^6 - 8*x^7 - 15*x^8 + ...
G.f. = q^-1 - q^2 - q^5 - 6*q^8 + 6*q^11 + 7*q^14 + 9*q^17 - 8*q^20 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)

Crossrefs

Cf. A030206.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3]^6 / QPochhammer[ x^9]^3, {x, 0, n}];

PARI

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

Formula

Expansion of q * eta(q^3) * eta(q^9)^6 / eta(q^27)^3 in powers of q^3.

Euler transform of period 9 sequence [ -1, -1, -7, -1, -1, -7, -1, -1, -4, ...].

a(5*n + 2) / a(2) == A030206(n) (mod 5). a(125*n + 42) / a(42) == A030206(n) (mod 25). [Amanda Clemm, 2016]