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Search: id:A000122
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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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19
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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; listen)
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OFFSET
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0,2
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COMMENT
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Theta series of the one-dimensional lattice Z: 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 + ...
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).
Euler transform of period 4 sequence [2,-3,2,-1,...].
Expansion of eta(q^2)^5/(eta(q)eta(q^4))^2 in powers of q.
G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=u^2-v^2+2w(w-u). - Michael Somos, Jul 20 2004
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REFERENCES
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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
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T. D. Noe, Table of n, a(n) for n=0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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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^(2k)) (1+x^(2k-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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MAPLE
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add(x^(m^2), m=-10..10);
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MATHEMATICA
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CoefficientList[ Sum[ x^(m^2), {m, -(n=10), n} ], x ]
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PROGRAM
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(PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(-X)^2/eta(X^2), n))
(PARI) a(n)=issquare(n)*2-(n==0)
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CROSSREFS
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Cf. A002448. Partial sums give A001650.
Sequence in context: A093492 A128771 A139380 this_sequence A002448 A033759 A033755
Adjacent sequences: A000119 A000120 A000121 this_sequence A000123 A000124 A000125
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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