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A000122
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Expansion of Jacobi theta function theta_3(x) = Sum_{m = -infinity..infinity} x^(m^2) (number of solutions to k^2 = n).
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595
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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 = -infinity..infinity} (-x)^(m^2).
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REFERENCES
| 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.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
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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.: Sum( x^(m^2), m=-infinity..infinity);
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).
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EXAMPLE
| 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 + 2*q^100 + ...
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MAPLE
| add(x^(m^2), m=-10..10);
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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 ]
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PROG
| (PARI) {a(n) = local(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 */
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CROSSREFS
| Cf. A002448. Partial sums give A001650.
Sequence in context: A128771 A139380 * A002448 A033759 A033755 A033753
Adjacent sequences: A000119 A000120 A000121 * A000123 A000124 A000125
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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