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A164273 Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.

%I A164273 #14 Feb 16 2025 08:33:11
%S A164273 1,-2,0,2,-2,0,0,4,0,-2,0,0,-2,-4,0,0,6,0,0,4,0,-4,0,0,0,-2,0,2,-4,0,
%T A164273 0,4,0,0,0,0,-2,-4,0,4,0,0,0,4,0,0,0,0,6,-6,0,0,-4,0,0,0,0,-4,0,0,0,
%U A164273 -4,0,4,6,0,0,4,0,0,0,0,0,-4,0,2,-4,0,0,4,0,-2,0,0,-4,0,0,0,0,0,0,8,0,-4,0,0,0,-4,0,0,-2,0,0,4,0
%N A164273 Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.
%C A164273 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A164273 G. C. Greubel, <a href="/A164273/b164273.txt">Table of n, a(n) for n = 0..1000</a>
%H A164273 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A164273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A164273 Expansion of eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
%F A164273 Euler transform of period 12 sequence [ -2, -1, 0, -1, -2, -4, -2, -1, 0, -1, -2, -2, ...].
%F A164273 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112605.
%F A164273 a(n) = (-1)^n * A164272(n).
%e A164273 G.f. = 1 - 2*q + 2*q^3 - 2*q^4 + 4*q^7 - 2*q^9 - 2*q^12 - 4*q^13 + 6*q^16 + ...
%t A164273 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0 ,q^3], {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%t A164273 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164273[n_] := SeriesCoefficient[f[-q, -q]*f[q^3, q^3], {q, 0, n}]; Table[A164273[n], {n, 0, 50}] (* _G. C. Greubel_, Sep 16 2017 *)
%o A164273 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
%Y A164273 Cf. A164272.
%K A164273 sign
%O A164273 0,2
%A A164273 _Michael Somos_, Aug 11 2009