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Search: id:A002448
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| A002448 |
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Expansion of Jacobi theta function theta_4(x). |
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+0
9
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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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Euler transform of period 2 sequence [ -2,-1,...].
Expansion of eta(q)^2/eta(q^2) in powers of q.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.11).
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, q-Series Identities
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FORMULA
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G.f.: Sum_n (-1)^n*x^(n^2) = Product_{n>0} (1-x^n)/(1+x^n).
a(n)=-2*b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos Jul 07 2006
a(3n+1)=-2*A089802(n), a(9n)=a(n). - Michael Somos Jul 07 2006
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MAPLE
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Sum((-x)^(m^2), m=-10..10);
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PROGRAM
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(PARI) a(n)=if(n<0, 0, (-1)^n*issquare(n)*2-(n==0))
(PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X)^2/eta(X^2), n))
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CROSSREFS
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Cf. A000122.
Sequence in context: A128771 A139380 A000122 this_sequence A033759 A033755 A033753
Adjacent sequences: A002445 A002446 A002447 this_sequence A002449 A002450 A002451
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KEYWORD
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sign
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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