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%I #41 Oct 26 2023 00:19:28
%S 1,-2,4,-8,14,-24,40,-64,100,-154,232,-344,504,-728,1040,-1472,2062,
%T -2864,3948,-5400,7336,-9904,13288,-17728,23528,-31066,40824,-53408,
%U 69568,-90248,116624,-150144,192612,-246256,313808
%N Expansion of 1 / Sum_{n=-oo..oo} x^(n^2).
%C Taylor series for 1/theta_3. Absolute values are coefficients in Taylor series for 1/theta_4.
%C Euler transform of period-4 sequence [-2,3,-2,1,...].
%D J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.
%H Robert Israel, <a href="/A004402/b004402.txt">Table of n, a(n) for n = 0..10000</a>
%H G. Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 103.
%F Ramanujan gave an asymptotic formula (see Almkvist).
%F G.f.: 1/Product_{m>0} ((1-q^(2m))(1+q^(2m-1))^2) = 1/theta_3(q).
%F a(n) = (-1)^n * A015128(n).
%p S:=series(1/JacobiTheta3(0,x),x,101):
%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Dec 29 2015
%t terms = 35; 1/EllipticTheta[3, 0, x] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Jul 05 2017 *)
%o (PARI) a(n)=if(n<0,0,polcoeff(1/sum(k=1,sqrtint(n),2*x^k^2,1+x*O(x^n)),n))
%o (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))}
%o (Julia) # JacobiTheta3 is defined in A000122.
%o A004402List(len) = JacobiTheta3(len, -1)
%o A004402List(35) |> println # _Peter Luschny_, Mar 12 2018
%Y See A015128 for a version without signs.
%K sign
%O 0,2
%A _N. J. A. Sloane_
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