A242405 - OEIS

%I #8 Mar 31 2018 07:33:02

%S 1,-6,15,-24,39,-72,123,-192,294,-456,693,-1008,1452,-2100,2991,-4176,

%T 5781,-7992,10950,-14808,19908,-26688,35541,-46944,61692,-80826,

%U 105366,-136536,176208,-226728,290565,-370704,471318,-597600,755217,-950976,1193988

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

%H G. C. Greubel, <a href="/A242405/b242405.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F 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.

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

%F Convolution square of A141094.

%F a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

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

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

%o (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))};

%Y Cf. A141094, A216046.

%K sign

%O 0,2

%A _Michael Somos_, May 13 2014