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A262056 Expansion of phi(q^2) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

%I A262056 #13 Feb 16 2025 08:33:27
%S A262056 1,0,2,2,0,4,4,0,10,8,0,20,14,0,36,24,0,64,42,0,108,68,0,176,108,0,
%T A262056 280,170,0,436,260,0,666,392,0,1000,584,0,1480,856,0,2160,1240,0,3116,
%U A262056 1780,0,4448,2526,0,6286,3552,0,8804,4956,0,12228,6856,0,16852
%N A262056 Expansion of phi(q^2) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
%C A262056 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262056 G. C. Greubel, <a href="/A262056/b262056.txt">Table of n, a(n) for n = 0..1000</a>
%H A262056 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262056 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262056 Expansion of eta(q^4)^5 * eta(q^6) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2) in powers of q.
%F A262056 Euler transform of period 24 sequence [ 0, 2, 2, -3, 0, 3, 0, -1, 2, 2, 0, -2, 0, 2, 2, -1, 0, 3, 0, -3, 2, 2, 0, 0, ...].
%F A262056 a(n) = A143068(2*n). a(3*n + 1) = 0.
%e A262056 G.f. = 1 + 2*q^2 + 2*q^3 + 4*q^5 + 4*q^6 + 10*q^8 + 8*q^9 + 20*q^11 + ...
%t A262056 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];
%o A262056 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2), n))};
%Y A262056 Cf. A143068.
%K A262056 nonn
%O A262056 0,3
%A A262056 _Michael Somos_, Sep 09 2015