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A143068 Expansion of phi(q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

%I A143068 #14 Jul 22 2025 06:10:59
%S A143068 1,2,0,0,2,0,2,4,0,2,4,0,4,8,0,4,10,0,8,16,0,8,20,0,14,30,0,16,36,0,
%T A143068 24,52,0,28,64,0,42,88,0,48,108,0,68,144,0,80,176,0,108,230,0,128,280,
%U A143068 0,170,360,0,200,436,0,260,552,0,308,666,0,392,832,0
%N A143068 Expansion of phi(q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.
%C A143068 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143068 G. C. Greubel, <a href="/A143068/b143068.txt">Table of n, a(n) for n = 0..1000</a>
%H A143068 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143068 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143068 Expansion of eta(q^2)^5 * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^2 in powers of q.
%F A143068 Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -1, 2, -1, 2, -3, 2, 0, ...].
%F A143068 G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143066.
%F A143068 a(3*n + 2) = 0.
%F A143068 G.f.: ( Sum_{k in Z} x^k^2 ) / ( Sum_{k in Z} (-x^6)^k^2 ).
%e A143068 G.f. = 1 + 2*q + 2*q^4 + 2*q^6 + 4*q^7 + 2*q^9 + 4*q^10 + 4*q^12 + 8*q^13 + ...
%t A143068 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o A143068 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};
%Y A143068 Cf. A143066.
%K A143068 nonn
%O A143068 0,2
%A A143068 _Michael Somos_, Jul 21 2008