A004402 Expansion of 1 / Sum_{n=-oo..oo} x^(n^2).

1, -2, 4, -8, 14, -24, 40, -64, 100, -154, 232, -344, 504, -728, 1040, -1472, 2062, -2864, 3948, -5400, 7336, -9904, 13288, -17728, 23528, -31066, 40824, -53408, 69568, -90248, 116624, -150144, 192612, -246256, 313808

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

See A015128 for a version without signs.

Sequence in context: A069252 A365667 A069253 * A015128 A208605 A123655

Adjacent sequences: A004399 A004400 A004401 * A004403 A004404 A004405

Keyword

sign

Author

N. J. A. Sloane

Status

approved