A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset

0,2

Comments

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

Theta series of the one-dimensional lattice Z.

Also, essentially the same as the theta series of the one-dimensional lattices A_1, A*_1, D_1, D*_1.

Number of ways of writing n as a square.

Closely related: theta_4(x) = Sum_{m = -oo..oo} (-x)^(m^2). See A002448.

Number 6 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

References

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, Exercise 1, p. 91.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 64.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5n].

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.1); p. 78, Eq. (32.22).

G. H. Hardy and E. M. Wright, Theorem 352, p. 282.

J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.

M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).

K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

Expansion of eta(q^2)^5 / (eta(q)*eta(q^4))^2 in powers of q.

Euler transform of period 4 sequence [2, -3, 2, -1, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v^2 + 2 * w * (w - u). - Michael Somos, Jul 20 2004

G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = w^4 - v^4 + w * (u - w)^3. - Michael Somos, May 11 2019

G.f.: Sum_{m=-oo..oo} x^(m^2);

a(0) = 1; for n > 0, a(n) = 0 unless n is a square when a(n) = 2.

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

G.f.: s(2)^5/(s(1)^2*s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

The Jacobi triple product identity states that for |x| < 1, z != 0, Product_{n>0} {(1-x^(2n))(1+x^(2n-1)z)(1+x^(2n-1)/z)} = Sum_{n=-inf..inf} x^(n^2)*z^n. Set z=1 to get theta_3(x).

For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1))). - Mikael Aaltonen, Jan 17 2015

G.f. is a period 1 Fourier series which satisfies f(-1/(4 t)) = 2^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016

a(n) = A000132(n)(mod 4). - John M. Campbell, Jul 07 2016

a(n) = (2/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

a(n) = 2 * A010052(n) if n>0. a(3*n + 1) = 2 * A089801(n). a(3*n + 2) = 0. a(4*n) = a(n). a(4*n + 2) = a(4*n + 3) = 0. a(8*n + 1) = 2 * A010054(n). - Michael Somos, May 11 2019

Dirichlet g.f.: 2*zeta(2s) - 1. - Francois Oger, Oct 26 2019

G.f. appears to equal exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021

From Peter Bala, Sep 27 2023: (Start)

G.f. A(x) satisfies A(x)*A(-x) = A(-x^2)^2.

A(x) = Sum_{n >= 1} x^(n-1)*Product_{k >= n} 1 - (-x)^k.

A(x)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*x^(2*n-1)/(1 - x^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.

A(x) = 1 + 2*Sum_{n >= 1} x^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + x^k ) /( Product_{k = 1..n} 1 + x^(2*k) ). See Fine, equation 14.43. (End)

Example

G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + ...

Maple

add(x^(m^2), m=-10..10): seq(coeff(%, x, n), n=0..100);
# alternative
A000122 := proc(n)
    if n = 0 then
        1;
    elif issqr(n) then
        2;
    else
        0 ;
    end if;
end proc:
seq(A000122(n), n=0..100) ; # R. J. Mathar, Feb 22 2021

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
CoefficientList[ Sum[ x^(m^2), {m, -(n=10), n} ], x ]
SquaresR[1, Range[0, 104]] (* Robert G. Wilson v, Jul 16 2014 *)
QP = QPochhammer; s = QP[q^2]^5/(QP[q]*QP[q^4])^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)
(4 QPochhammer[q^2]/QPochhammer[-1, -q]^2 + O[q]^101)[[3]] (* Vladimir Reshetnikov, Sep 16 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 14 2011 */
(PARI) {a(n) = issquare(n) * 2 -(n==0)}; /* Michael Somos, Jun 17 1999 */
(Magma) Basis( ModularForms( Gamma0(4), 1/2), 100) [1]; /* Michael Somos, Jun 10 2014 */
(Magma) L := Lattice("A", 1); A<q> := ThetaSeries(L, 20); A; /* Michael Somos, Nov 13 2014 */
(Sage)
Q = DiagonalQuadraticForm(ZZ, [1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014
(Julia)
using Nemo
function JacobiTheta3(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A000122List(len) = JacobiTheta3(len, 1)
A000122List(105) |> println # Peter Luschny, Mar 12 2018
(Python)
from sympy.ntheory.primetest import is_square
def A000122(n): return is_square(n)<<1 if n else 1 # Chai Wah Wu, May 17 2023

Crossrefs

1st column of A286815. - Seiichi Manyama, May 27 2017

Row d=1 of A122141.

Cf. A002448 (theta_4). Partial sums give A001650.

Cf. A010052, A010054, A089801.

Cf. A000007, A004015, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_3, A_2, A_4, ...).

Sequence in context: A093492 A139380 A128771 * A002448 A033759 A033755

Adjacent sequences: A000119 A000120 A000121 * A000123 A000124 A000125

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane

Status

approved