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%I #19 Sep 27 2019 13:42:49
%S 1,2,3,8,13,20,37,56,83,134,196,280,419,592,824,1176,1618,2202,3040,
%T 4096,5471,7368,9753,12824,16937,22090,28653,37248,47968,61488,78887,
%U 100472,127461,161702,203951,256368,322090,402748,502112,625464,776061
%N Expansion of (b(q^6) * c(q^6)) / (b(q^3) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.
%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).
%C Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 4*u*v^2.
%C Also, B(q) satisfies 0 = f(B(q), B(-q)) where f(u, v) = u + v - 4*u^2*v^2 which is involved in equation (13.22) where gg' = (2B(q))^12 and GG' = (2B(-q))^12. Refer to A058092 for more details. - _Michael Somos_, Sep 27 2019
%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag. See p. 179, equation (13.22).
%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.
%H G. C. Greubel, <a href="/A102315/b102315.txt">Table of n, a(n) for n = 0..1000</a>
%F Expansion of (chi(-x) * chi(-x^3))^(-2) in powers of x where chi() is a Ramanujan theta function.
%F Euler transform of period 6 sequence [2, 0, 4, 0, 2, 0, ...].
%F Expansion of q^(-1) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^2 in powers of q^3.
%F Convolution inverse of A058543. - _Michael Somos_, Feb 19 2015
%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 08 2015
%e G.f. = 1 + 2*x + 3*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 37*x^6 + 56*x^7 + ...
%e G.f. = q + 2*q^4 + 3*q^7 + 8*q^10 + 13*q^13 + 20*q^16 + 37*q^19 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6])^-2, {x, 0, n}]; (* _Michael Somos_, Feb 19 2015 *)
%t nmax = 60; CoefficientList[Series[Product[(1+x^k)^2 * (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 08 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^2, n))};
%o (PARI) q='q+O('q^99); Vec((eta(q^2)*eta(q^6)/(eta(q)*eta(q^3)))^2) \\ _Altug Alkan_, Apr 21 2018
%Y Cf. A058092, A058543.
%K nonn
%O 0,2
%A _Michael Somos_, Jan 04 2005
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