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

Author

N. J. A. Sloane

Keywords

• sign

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.

Crossrefs

See A015128 for a version without signs.

Programs

Julia

# JacobiTheta3 is defined in A000122.
A004402List(len) = JacobiTheta3(len, -1)
A004402List(35) |> println # Peter Luschny, Mar 12 2018

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 *)

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))}

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).