A242405 Expansion of (b(q) / b(q^2))^2 in powers of q where b() is a cubic AGM theta function.

1, -6, 15, -24, 39, -72, 123, -192, 294, -456, 693, -1008, 1452, -2100, 2991, -4176, 5781, -7992, 10950, -14808, 19908, -26688, 35541, -46944, 61692, -80826, 105366, -136536, 176208, -226728, 290565, -370704, 471318, -597600, 755217, -950976, 1193988

Offset: 0

Author

Michael Somos, May 13 2014

Keywords

• sign

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).

Examples

			G.f. = 1 - 6*q + 15*q^2 - 24*q^3 + 39*q^4 - 72*q^5 + 123*q^6 - 192*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A141094, A216046.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2]^3 / QPochhammer[ x^3, x^6])^2, {x, 0, n}];

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)))^2, n))};

Formula

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

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

Euler transform of period 6 sequence [ -6, 0, -4, 0, -6, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A216046.

G.f.: Product_{k>0} ((1 - x^(2*k-1))^3 / (1 - x^(6*k-3)))^2.

Convolution square of A141094.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017