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A261576 Expansion of 3 * b(q^2) * c(q^2) / c(q)^2 in powers of q where b(), c() are cubic AGM theta functions.

%I A261576 #10 Feb 16 2025 08:33:26
%S A261576 1,-2,-3,12,-10,-18,60,-48,-75,228,-172,-252,732,-524,-744,2088,-1450,
%T A261576 -1998,5460,-3704,-4986,13344,-8872,-11736,30876,-20206,-26322,68268,
%U A261576 -44080,-56682,145224,-92672,-117867,298800,-188756,-237744,597108,-373852,-466836
%N A261576 Expansion of 3 * b(q^2) * c(q^2) / c(q)^2 in powers of q where b(), c() are cubic AGM theta functions.
%C A261576 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A261576 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261576 G. C. Greubel, <a href="/A261576/b261576.txt">Table of n, a(n) for n = 0..1000</a>
%H A261576 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261576 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261576 Expansion of f(-q^2)^4 / (f(-q^3)^2 * f(q, q^2)^2) in powers of q where f(,) is Ramanujan's general theta function.
%F A261576 Expansion of (eta(q) * eta(q^2) * eta(q^6) / eta(q^3)^3)^2 in powers of q.
%F A261576 Euler transform of period 6 sequence [ -2, -4, 4, -4, -2, 0, ...].
%F A261576 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 9/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A258099.
%F A261576 a(n) = A261326(2*n).  a(3*n + 2) = -3 * A233698(n).
%e A261576 G.f. = 1 - 2*x - 3*x^2 + 12*x^3 - 10*x^4 - 18*x^5 + 60*x^6 - 48*x^7 + ...
%t A261576 a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6] / QPochhammer[ q^3]^3)^2, {q, 0, n}];
%o A261576 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A)^3)^2, n))};
%Y A261576 Cf. A233698, A258099, A261326.
%K A261576 sign
%O A261576 0,2
%A A261576 _Michael Somos_, Aug 25 2015