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A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.

%I A118272 #17 Feb 16 2025 08:33:01
%S A118272 1,-2,1,-4,8,-6,6,-8,14,-10,1,-16,20,-14,12,-16,31,-18,8,-20,32,-28,
%T A118272 18,-24,38,-32,6,-28,44,-30,24,-40,57,-34,14,-36,72,-38,30,-48,62,-52,
%U A118272 1,-44,68,-46,48,-56,74,-50,20,-64,80,-64,42,-56,108,-58,12,-60,112,-76,48,-64,98,-66,31,-80,104,-80,54,-88
%N A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.
%C A118272 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A118272 G. C. Greubel, <a href="/A118272/b118272.txt">Table of n, a(n) for n = 0..1000</a>
%H A118272 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A118272 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A118272 Expansion of f(-x^3)^6 / f(x, x^2)^2 = phi(-x^3)^2 * f(-x, -x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions. - _Michael Somos_, Mar 22 2015
%F A118272 Euler transform of period 6 sequence [ -2, 0, -4, 0, -2, -4, ...].
%F A118272 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 16 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252651. - _Michael Somos_, Mar 22 2015
%F A118272 G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2. - _Michael Somos_, Mar 22 2015
%F A118272 -3 * a(n) = A118271(3*n + 2).
%e A118272 G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 6*x^6 - 8*x^7 + 14*x^8 + ...
%e A118272 G.f. = q^2 - 2*q^5 + q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 6*q^20 - 8*q^23 + ...
%t A118272 QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[x^3]^6/(QP[-x, x^3]* QP[-x^2, x^3]*QP[x^3])^2, {x, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Apr 15 2018 *)
%o A118272 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A))^2, n))};
%o A118272 (PARI) q='q+O('q^99); Vec((eta(q)*eta(q^3)*eta(q^6)/eta(q^2))^2) \\ _Altug Alkan_, Apr 16 2018
%o A118272 (Magma) A := Basis( ModularForms( Gamma0(36), 2), 180); A[3] - 2*A[6] + A[9]; /* _Michael Somos_, Mar 22 2015 */
%Y A118272 Cf. A118271, A252651.
%K A118272 sign
%O A118272 0,2
%A A118272 _Michael Somos_, Apr 21 2006