OFFSET
0,2
COMMENTS
Taylor series for 1/theta_3. Absolute values are coefficients in Taylor series for 1/theta_4.
Euler transform of period-4 sequence [-2,3,-2,1,...].
REFERENCES
J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.
FORMULA
Ramanujan gave an asymptotic formula (see Almkvist).
G.f.: 1/Product_{m>0} ((1-q^(2m))(1+q^(2m-1))^2) = 1/theta_3(q).
a(n) = (-1)^n * A015128(n).
MAPLE
S:=series(1/JacobiTheta3(0, x), x, 101):
seq(coeff(S, x, j), j=0..100); # Robert Israel, Dec 29 2015
MATHEMATICA
terms = 35; 1/EllipticTheta[3, 0, x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Jul 05 2017 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(1/sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)), n))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2/eta(x^2+A)^5, n))}
(Julia) # JacobiTheta3 is defined in A000122.
A004402List(len) = JacobiTheta3(len, -1)
A004402List(35) |> println # Peter Luschny, Mar 12 2018
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved