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A121590 Expansion of (eta(q^3) / eta(q))^12 in powers of q.

%I A121590 #28 Oct 18 2018 03:06:10
%S A121590 1,12,90,508,2391,9828,36428,124188,395199,1186344,3387252,9257364,
%T A121590 24343037,61848096,152356032,364959196,852243948,1944226476,
%U A121590 4341094220,9502198728,20419293123,43131708720,89656112256,183580652340
%N A121590 Expansion of (eta(q^3) / eta(q))^12 in powers of q.
%C A121590 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A121590 Seiichi Manyama, <a href="/A121590/b121590.txt">Table of n, a(n) for n = 1..10000</a>
%H A121590 Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
%F A121590 Expansion of (c(q) / (3 * b(q)))^3 in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Jun 16 2012
%F A121590 Euler transform of period 3 sequence [ 12, 12, 0, ...].
%F A121590 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - u*v - 24*u*v * (u + v) - 729*u^2*v^2.
%F A121590 G.f.: x * (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^12.
%F A121590 Convolution inverse of A030182. - _Michael Somos_, Jun 16 2012
%F A121590 G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^-6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030182.
%F A121590 Convolution 12th power of A000726, cube of A128758, square of A121596. - _Michael Somos_, Aug 09 2015
%F A121590 a(n) ~ exp(4*Pi*sqrt(n/3)) / (729 * sqrt(2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%F A121590 a(1) = 1, a(n) = (12/(n-1))*Sum_{k=1..n-1} A046913(k)*a(n-k) for n > 1. - _Seiichi Manyama_, Apr 01 2017
%e A121590 G.f. = q + 12*q^2 + 90*q^3 + 508*q^4 + 2391*q^5 + 9828*q^6 + 36428*q^7 + ...
%t A121590 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] / QPochhammer[ q])^12, {q, 0, n}]; (* _Michael Somos_, Aug 09 2015 *)
%t A121590 nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(3*k)) / (1 - x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%o A121590 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A))^12, n))};
%Y A121590 Cf. A000726, A030182, A128758.
%K A121590 nonn
%O A121590 1,2
%A A121590 _Michael Somos_, Aug 09 2006