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%I #12 Mar 12 2021 22:24:48
%S 1,2,1,2,4,4,2,0,6,6,1,4,6,8,2,0,7,6,4,6,8,8,4,0,10,8,2,6,10,12,0,0,9,
%T 14,6,6,12,8,6,0,10,12,1,10,14,8,4,0,16,14,6,8,8,16,8,0,12,14,2,10,12,
%U 16,0,0,20,10,7,8,20,20,6,0,10,16,4,10,20,12
%N Expansion of chi(x)^2 * f(-x^6)^3 in powers of x where chi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A257651/b257651.txt">Table of n, a(n) for n = 0..2500</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^3) * f(x, x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%F Expansion of phi(x) * c(x^2) / 3 in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%F Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6)^3 / (eta(q)^2 * eta(q^4)^2) in powers of q.
%F Euler transform of period 12 sequence [ 2, -2, 2, 0, 2, -5, 2, 0, 2, -2, 2, -3, ...].
%F a(n) = a(4*n + 2) = A213592(2*n + 1). a(2*n) = A213592(n). a(2*n + 1) = 2 * A213607(n).
%F a(8*n + 2) = A213592(n). a(8*n + 3) = 2 * A213617(n). a(8*n + 5) = 4 * A213023(n). a(8*n + 6) = 2 * A213607(n). a(8*n + 7) = 0.
%e G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 2*x^6 + 6*x^8 + 6*x^9 + ...
%e G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 4*q^14 + 4*q^17 + 2*q^20 + 6*q^26 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2 QPochhammer[ x^6]^3, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2), n))};
%Y Cf. A213023, A213592, A213607, A213617.
%K nonn
%O 0,2
%A _Michael Somos_, Jul 25 2015
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