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%I #22 Mar 12 2021 22:24:45
%S 1,2,-1,-2,1,0,2,-2,-6,4,-4,-6,1,-4,6,4,0,2,2,4,6,10,-1,6,-3,-12,-6,0,
%T 8,-12,2,-2,-2,-2,-12,12,2,2,0,-8,-11,-6,6,12,-6,-4,8,-4,2,0,6,-14,4,
%U 6,2,4,-6,6,2,12,-11,12,-1,-2,20,0,-8,4,18,4,12,0,-6,-6,-6,-20,-6,-4,-22,-12,12,10,0,-18,-9,4,-6,-2,-24
%N Coefficients of L-series for elliptic curve "80b2": y^2 = x^3 - x^2 - x.
%C Number 61 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Seiichi Manyama, <a href="/A159817/b159817.txt">Table of n, a(n) for n = 0..1000</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (f(x) * f(x^5))^2 in powers of x where f() is a Ramanujan theta function.
%F Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)))^2 in powers of q.
%F Euler transform of period 20 sequence [ 2, -4, 2, -2, 4, -4, 2, -2, 2, -8, 2, -2, 2, -4, 4, -2, 2, -4, 2, -4, ...].
%F a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 80 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F G.f.: (Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)))^2.
%F a(n) = (-1)^n * A030205(n). Convolution square of A159818.
%e G.f. = 1 + 2*x - x^2 - 2*x^3 + x^4 + 2*x^6 - 2*x^7 - 6*x^8 + 4*x^9 - 4*x^10 + ...
%e G.f. = q + 2*q^3 - q^5 - 2*q^7 + q^9 + 2*q^13 - 2*q^15 - 6*q^17 + 4*q^19 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] QPochhammer[ -x^5])^2, {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o (PARI) {a(n) = if( n<0, 0, ellak( ellinit([0, -1, 0, -1, 0], 1), 2*n + 1))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^10 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)))^2, n))};
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==5, (-1)^e, a0=1; a1 = y = -sum(x=0, p-1, kronecker(x^3 - x^2 - x, p)); for(i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))};
%Y Cf. A030205, A159818.
%K sign
%O 0,2
%A _Michael Somos_, Apr 22 2009
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