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Expansion of (b(q^3)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM theta function.
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%I #14 Aug 11 2018 21:57:17

%S 1,-3,0,13,-24,0,50,-51,0,72,-120,0,170,-150,0,205,-288,0,362,-312,0,

%T 360,-528,0,601,-510,0,650,-840,0,962,-819,0,864,-1200,0,1370,-1086,0,

%U 1224,-1680,0,1850,-1560,0,1584,-2208,0,2451,-1803,0,2210,-2808,0

%N Expansion of (b(q^3)^3 - b(q)^3) / 9 in powers of q where b() is a cubic AGM theta function.

%H Andrew Howroyd, <a href="/A181905/b181905.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F a(3*n) = 0.

%F Conjecture: multiplicative with a(3^e) = 0, a(p^e) = ((p^2)^(e+1)-1)/(p^2-1) for p == 1 (mod 3), a(p^e) = (1-(-p^2)^(e+1))/(p^2+1) for p == 2 (mod 3). - _Andrew Howroyd_, Aug 05 2018

%e q - 3*q^2 + 13*q^4 - 24*q^5 + 50*q^7 - 51*q^8 + 72*q^10 - 120*q^11 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[ ((eta[q^3]^3/ eta[q^9])^3 - (eta[q]^3/eta[q^3])^3)/9, {q, 0, 50}], q]] (* _G. C. Greubel_, Aug 11 2018 *)

%o (PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( ((eta(x^3 + A)^3 / eta(x^9 + A))^3 - (eta(x + A)^3 / eta(x^3 + A))^3) / 9, n))}

%K sign

%O 1,2

%A _Michael Somos_, Apr 01 2012