A159817 Coefficients of L-series for elliptic curve "80b2": y^2 = x^3 - x^2 - x.

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, 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, 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

Offset

0,2

Comments

Number 61 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (f(x) * f(x^5))^2 in powers of x where f() is a Ramanujan theta function.

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.

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, ...].

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.

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).

G.f.: (Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)))^2.

a(n) = (-1)^n * A030205(n). Convolution square of A159818.

Example

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 + ...
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 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] QPochhammer[ -x^5])^2, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, ellak( ellinit([0, -1, 0, -1, 0], 1), 2*n + 1))};
(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))};
(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)))};

Crossrefs

Cf. A030205, A159818.

Sequence in context: A367169 A236831 A030205 * A079532 A328176 A191312

Adjacent sequences: A159814 A159815 A159816 * A159818 A159819 A159820

Keyword

sign

Author

Michael Somos, Apr 22 2009

Status

approved