%I #11 Mar 12 2021 22:24:48
%S 1,3,4,5,6,8,9,6,8,8,9,12,8,12,8,13,18,8,16,12,16,13,6,16,8,18,20,16,
%T 15,12,24,18,16,16,12,20,17,18,16,16,30,24,16,12,16,21,24,16,16,18,20,
%U 32,18,28,24,27,20,16,24,12,32,30,24,16,18,32,25,18,32
%N Expansion of f(x, x) * f(x, x^2) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A298420/b298420.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 phi(-x^2) * phi(-x^3) * phi(-x^6) / chi(-x)^3 in powers of x where phi(), chi() are Ramanujan theta functions.
%F Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^6) / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [3, -2, 1, -1, 3, -5, 3, -1, 1, -2, 3, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 48^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298421.
%e G.f. = 1 + 3*x + 4*x^2 + 5*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + ...
%e G.f. = q + 3*q^9 + 4*q^17 + 5*q^25 + 6*q^33 + 8*q^41 + 9*q^49 + 6*q^57 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q, q]^2 QPochhammer[ q^3]^2 QPochhammer[ q^6, q^12], {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x, x]^3, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};
%Y Cf. A298421.
%K nonn
%O 0,2
%A _Michael Somos_, Jan 18 2018