A215711 - OEIS

%I #15 Aug 10 2018 16:23:07

%S 1,-3,-27,159,-219,-378,1431,-1032,-1755,4533,-3402,-3996,11607,-6594,

%T -9288,20034,-14043,-14742,40797,-20580,-27594,54696,-35964,-36504,

%U 93015,-47253,-59346,122631,-75336,-73170,180306,-89376,-112347,211788,-132678,-130032

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

%H G. C. Greubel, <a href="/A215711/b215711.txt">Table of n, a(n) for n = 0..2500</a>

%H R. S. Maier, <a href="http://arXiv.org/abs/0807.1081">Nonlinear differential equations satisfied by certain classical modular forms</a>, arXiv:0807.1081 [math.NT], 2008-2010. See Table 1, p.12

%F Expansion of (1 + 9 * q * (eta(q^9) / eta(q))^3) * (eta(q)^3 / eta(q^3))^4 in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A198956.

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

%F Convolution of A215690 and A133078. Convolution of A004016 and A109041.

%e G.f. = 1 - 3*q - 27*q^2 + 159*q^3 - 219*q^4 - 378*q^5 + 1431*q^6 - 1032*q^7 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(1 + 9*(eta[q^9]/eta[q])^3)*(eta[q]^3/eta[q^3])^4, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 10 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3) * (eta(x + A)^3 / eta(x^3 + A))^4, n))}

%Y Cf. A004016, A109041, A133078, A198956, A215690.

%K sign

%O 0,2

%A _Michael Somos_, Aug 21 2012