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A263773 Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.

%I A263773 #15 Feb 16 2025 08:33:27
%S A263773 1,6,9,-12,-42,-18,36,48,45,-12,-108,-36,84,84,72,-72,-186,-54,36,120,
%T A263773 126,-96,-216,-72,180,186,126,-12,-336,-90,216,192,189,-144,-324,-144,
%U A263773 84,228,180,-168,-540,-126,288,264,252,-72,-432,-144,372,342,279,-216
%N A263773 Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.
%C A263773 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A263773 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263773 G. C. Greubel, <a href="/A263773/b263773.txt">Table of n, a(n) for n = 0..1000</a>
%H A263773 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263773 Expansion of f(q)^6 / f(q^3)^2 in powers of q where f() is a Ramanujan theta function.
%F A263773 Expansion of (eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^3))^2 in powers of q.
%F A263773 Euler transform of period 12 sequence [ 6, -12, 4, -6, 6, -8, 6, -6, 4, -12, 6, -4, ...].
%F A263773 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134079.
%F A263773 G.f.: Product_{k>0} (1 - (-x)^k)^6 / (1 - (-x)^(3*k))^2.
%F A263773 a(2*n + 1) = 6 * A252651(n). a(3*n + 2) = 9 * A134079(n).
%F A263773 Convolution square of A226535.
%e A263773 G.f. = 1 + 6*x + 9*x^2 - 12*x^3 - 42*x^4 - 18*x^5 + 36*x^6 + 48*x^7 + 45*x^8 + ...
%t A263773 a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^6 / QPochhammer[ -q^3]^2, {q, 0, n}];
%o A263773 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^3))^2, n))};
%Y A263773 Cf. A134079, A226535, A262651.
%K A263773 sign
%O A263773 0,2
%A A263773 _Michael Somos_, Oct 27 2015