%I #13 Sep 08 2022 08:46:09

%S 1,-2,-2,2,6,4,-14,0,6,-2,-12,-8,42,4,-16,-4,6,-4,-50,8,36,0,-24,16,

%T 42,2,-28,2,48,-12,-84,-16,6,8,-36,0,150,-12,-40,-4,36,12,-112,-8,72,

%U 4,-48,0,42,14,-62,4,84,4,-158,16,48,-8,-60,-8,252,4,-64,0,6

%N Expansion of phi(-q) * phi(-q^2) * phi(-q^3) * phi(-q^6) in powers of q.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A246707/b246707.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^6) / (eta(q^4) * eta(q^12)) in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 384 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033765.

%F a(2*n + 1) = -2 * A030188(n).

%e G.f. = 1 - 2*q - 2*q^2 + 2*q^3 + 6*q^4 + 4*q^5 - 14*q^6 + 6*q^8 - 2*q^9 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^2* eta[q^2]*eta[q^3]^2*eta[q^6]/(eta[q^4]*eta[q^12]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Apr 18 2018 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x^4 + A) * eta(x^12 + A)), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)*eta(q^3)^2*eta(q^6)/(eta(q^4)*eta(q^12))) \\ _Altug Alkan_, Apr 18 2018

%o (Magma) A := Basis( ModularForms( Gamma0(24), 2), 26); A[1] - 2*A[2] - 2*A[3] + 2*A[4] + 6*A[5] + 4*A[6] - 14*A[7] + 6*A[8];

%Y Cf. A030188, A033765.

%K sign

%O 0,2

%A _Michael Somos_, Sep 01 2014