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A260309 Expansion of phi(q) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

%I A260309 #11 Feb 16 2025 08:33:26
%S A260309 1,2,0,0,2,0,-2,-4,0,2,-4,0,0,0,0,-4,2,0,0,0,0,0,-4,0,2,6,0,0,4,0,0,
%T A260309 -4,0,4,0,0,2,0,0,0,4,0,-4,0,0,0,0,0,0,6,0,0,0,0,-2,-8,0,0,-4,0,4,0,0,
%U A260309 -4,2,0,0,0,0,0,-8,0,0,4,0,0,0,0,0,-4,0,2
%N A260309 Expansion of phi(q) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.
%C A260309 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260309 G. C. Greubel, <a href="/A260309/b260309.txt">Table of n, a(n) for n = 0..1000</a>
%H A260309 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260309 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260309 Expansion of eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^4)^2* eta(q^12)) in powers of q.
%F A260309 Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -5, 2, -1, 2, -3, 2, -2, ...].
%F A260309 G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 1536^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260308.
%F A260309 G.f.: Sum_{i,j in Z} (-1)^j * x^(i^2 + 6*j^2).
%F A260309 a(n) = (-1)^floor(n/2) * A046113(n). a(3*n + 2) = 0. a(4*n) = A046113(n). 2 * a(n) = A260110(n).
%e A260309 G.f. = 1 + 2*x + 2*x^4 - 2*x^6 - 4*x^7 + 2*x^9 - 4*x^10 - 4*x^15 + ...
%t A260309 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, -q^6], {q, 0, n}];
%o A260309 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2* eta(x^12 + A)), n))};
%Y A260309 Cf. A046113, A260110, A260308.
%K A260309 sign
%O A260309 0,2
%A A260309 _Michael Somos_, Jul 22 2015