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A233670 Expansion of q * phi(-q^2) * psi(q^9) / (f(q^3) * phi(q^3)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.

%I A233670 #14 Feb 16 2025 08:33:21
%S A233670 1,0,-2,-3,0,6,8,0,-14,-18,0,30,38,0,-60,-75,0,114,140,0,-208,-252,0,
%T A233670 366,439,0,-626,-744,0,1044,1232,0,-1704,-1998,0,2730,3182,0,-4300,
%U A233670 -4986,0,6672,7700,0,-10212,-11736,0,15438,17673,0,-23076,-26322,0,34134
%N A233670 Expansion of q * phi(-q^2) * psi(q^9) / (f(q^3) * phi(q^3)) in powers of q where f(), phi(), psi() are Ramanujan theta functions.
%C A233670 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A233670 G. C. Greubel, <a href="/A233670/b233670.txt">Table of n, a(n) for n = 1..1000</a>
%H A233670 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A233670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A233670 Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2 / (eta(q^4) * eta(q^6)^8 * eta(q^9)) in powers of q.
%F A233670 Euler transform of period 36 sequence [ 0, -2, -3, -1, 0, 3, 0, -1, -2, -2, 0, 1, 0, -2, -3, -1, 0, 2, 0, -1, -3, -2, 0, 1, 0, -2, -2, -1, 0, 3, 0, -1, -3, -2, 0, 0, ...].
%F A233670 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
%F A233670 a(3*n) = -2 * A228447(n). a(6*n) = 6 * A128638(n). A(3*n + 2) = 0.
%F A233670 Convolution inverse of A058647.
%e A233670 G.f. = q - 2*q^3 - 3*q^4 + 6*q^6 + 8*q^7 - 14*q^9 - 18*q^10 + 30*q^12 + ...
%t A233670 A233670[n_] := SeriesCoefficient[q*QPochhammer[q^2]^2*QPochhammer[q^3]^3 *QPochhammer[q^12]^3*QPochhammer[q^18]^2/(QPochhammer[q^4] * QPochhammer[q^6]^8*QPochhammer[q^9]), {q, 0, n}]; Table[A233670[n], {n, 50}] (* _G. C. Greubel_, Oct 09 2017 *)
%o A233670 (PARI) {a(n) = my(A); if( n<1, 0, n--; 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^4 + A) * eta(x^6 + A)^8 * eta(x^9 + A)), n))};
%Y A233670 Cf. A058647, A128638, A228447.
%K A233670 sign
%O A233670 1,3
%A A233670 _Michael Somos_, Dec 14 2013