A231962 - OEIS

%I #9 Aug 08 2018 22:54:32

%S 1,-18,0,-90,-234,0,-216,-900,0,-738,-1296,0,-1170,-3060,0,-1728,

%T -3690,0,-2160,-6516,0,-4500,-6480,0,-3672,-10818,0,-6570,-11700,0,

%U -6480,-17316,0,-8640,-15552,0,-9594,-24660,0,-15300,-22032,0,-10800,-33300,0,-17280

%N Expansion of b(q)^3 - (1/3)*c(q)^3 in powers of q where b(), c() are cubic AGM theta functions.

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

%H G. C. Greubel, <a href="/A231962/b231962.txt">Table of n, a(n) for n = 0..2500</a>

%F Expansion of (eta(q)^3 / eta(q^3))^3 - 9 * q * (eta(q^3)^3 / eta(q))^3 in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = - 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231961.

%F a(3*n + 2) = 0. a(3*n + 1) = -18 * A231947(n). a(3*n) = A231961(n).

%e G.f. = 1 - 18*q - 90*q^3 - 234*q^4 - 216*q^6 - 900*q^7 - 738*q^9 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^3/ eta[q^3])^3 - 9*(eta[q^3]^3/eta[q])^3, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 08 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 - 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}

%Y Cf. A231947, A231961.

%K sign

%O 0,2

%A _Michael Somos_, Nov 15 2013