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A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.

%I A233672 #19 Feb 16 2025 08:33:21
%S A233672 1,1,0,-2,-3,0,6,8,0,-14,-18,0,30,38,0,-60,-75,0,114,140,0,-208,-252,
%T A233672 0,366,439,0,-626,-744,0,1044,1232,0,-1704,-1998,0,2730,3182,0,-4300,
%U A233672 -4986,0,6672,7700,0,-10212,-11736,0,15438,17673,0,-23076,-26322,0
%N A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%C A233672 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A233672 G. C. Greubel, <a href="/A233672/b233672.txt">Table of n, a(n) for n = 0..1000</a>
%H A233672 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A233672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A233672 Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2 / (eta(q) * eta(q^6)^8 * eta(q^36)) in powers of q.
%F A233672 Euler transform of period 36 sequence [ 1, -1, -2, -1, 1, 4, 1, -1, -2, -1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, -1, -2, -1, 1, 4, 1, -1, -2, -1, 1, 0, ...].
%F A233672 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F A233672 a(n) = A233670(n) unless n=0.
%e A233672 G.f. = 1 + q - 2*q^3 - 3*q^4 + 6*q^6 + 8*q^7 - 14*q^9 - 18*q^10 + ...
%t A233672 eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q^2]^2 *eta[q^3]^3*eta[q^12]^3*eta[q^18]^2/(eta[q]*eta[q^6]^8*eta[q^36]), {q, 0, 50}], q] (* _G. C. Greubel_, Aug 08 2018 *)
%o A233672 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 * eta(x^18 + A)^2 / (eta(x + A) * eta(x^6 + A)^8 * eta(x^36 + A)), n))};
%Y A233672 Cf. A233670.
%K A233672 sign
%O A233672 0,4
%A A233672 _Michael Somos_, Dec 14 2013