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A138518 Expansion of (phi(-q) / phi(-q^5))^2 in powers of q where phi() is a Ramanujan theta function.

%I A138518 #15 Feb 16 2025 08:33:08
%S A138518 1,-4,4,0,4,-4,-16,16,4,12,-12,-48,48,8,32,-32,-124,120,20,80,-76,
%T A138518 -288,272,48,176,-164,-616,576,96,360,-336,-1248,1156,192,712,-656,
%U A138518 -2412,2216,368,1344,-1228,-4488,4096,672,2448,-2228,-8096,7344,1200,4348,-3932
%N A138518 Expansion of (phi(-q) / phi(-q^5))^2 in powers of q where phi() is a Ramanujan theta function.
%C A138518 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138518 G. C. Greubel, <a href="/A138518/b138518.txt">Table of n, a(n) for n = 0..1000</a>
%H A138518 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138518 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138518 Expansion of ( (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) )^2 in powers of q.
%F A138518 Euler transform of period 10 sequence [ -4, -2, -4, -2, 0, -2, -4, -2, -4, 0, ...].
%F A138518 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u) * (u - 1) - 4 * u * (v - 1).
%F A138518 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u - 5) * v * (v - 1) * (v - 5).
%F A138518 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138519.
%F A138518 G.f.: (Product_{k>0} P(10, x^k) / P(5, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.
%F A138518 a(n) = -4 * A095813(n) unless n=0.
%F A138518 Convolution inverse of A138517. Convolution square of A138527.
%e A138518 G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 4*q^5 - 16*q^6 + 16*q^7 + 4*q^8 + 12*q^9 + ...
%t A138518 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5])^2, {q, 0, n}]; (* _Michael Somos_, Sep 16 2015 *)
%o A138518 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A))^2, n))};
%Y A138518 Cf. A095813, A138517, A138519, A138527.
%K A138518 sign
%O A138518 0,2
%A A138518 _Michael Somos_, Mar 23 2008