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A139213 Expansion of phi(q) * phi(-q^18) / (phi(-q^3) * phi(-q^6)) in powers of q where phi() is a Ramanujan theta function.

%I A139213 #13 Feb 16 2025 08:33:08
%S A139213 1,2,0,2,6,0,6,16,0,14,36,0,30,76,0,60,150,0,114,280,0,208,504,0,366,
%T A139213 878,0,626,1488,0,1044,2464,0,1704,3996,0,2730,6364,0,4300,9972,0,
%U A139213 6672,15400,0,10212,23472,0,15438,35346,0,23076,52644,0,34134,77616,0
%N A139213 Expansion of phi(q) * phi(-q^18) / (phi(-q^3) * phi(-q^6)) in powers of q where phi() is a Ramanujan theta function.
%C A139213 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A139213 G. C. Greubel, <a href="/A139213/b139213.txt">Table of n, a(n) for n = 0..1000</a>
%H A139213 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A139213 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A139213 Expansion of eta(q^2)^5 * eta(q^12) * eta(q^18)^2 / (eta(q)^2 * eta(q^3)^2 * eta(q^4)^2 * eta(q^6) * eta(q^36)) in powers of q.
%F A139213 Euler transform of period 36 sequence [ 2, -3, 4, -1, 2, 0, 2, -1, 4, -3, 2, 1, 2, -3, 4, -1, 2, -2, 2, -1, 4, -3, 2, 1, 2, -3, 4, -1, 2, 0, 2, -1, 4, -3, 2, 0, ...].
%F A139213 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139215.
%F A139213 a(n) = 2 * A139214(n) unless n=0. a(3*n + 2) = 0.
%e A139213 G.f. = 1 + 2*q + 2*q^3 + 6*q^4 + 6*q^6 + 16*q^7 + 14*q^9 + 36*q^10 + ...
%t A139213 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^18] / (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^6]), {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o A139213 (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^18 + A)^2 / (eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^36 + A)), n))};
%Y A139213 Cf. A139214, A139215.
%K A139213 nonn
%O A139213 0,2
%A A139213 _Michael Somos_, Apr 11 2008