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A089807
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Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.
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2
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1, -1, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,10
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| I. J. Zucker, "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.
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LINKS
| 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
| a(n) = -b(n) where b(n) is multiplicative and b(3^e) = -2(1+(-1)^e)/2 if e>0, b(p^e) = (1+(-1)^e)/2 otherwise.
Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26, 2006
Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Nov 05 2005
Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f() is Ramanujan's two variable theta function. - Michael Somos, Sep 17 2007
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...]. - Michael Somos, Nov 05 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A089801.
G.f.: (Sum_{k} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))) - Michael Somos, Nov 05 2005
a(n) = (-1)^n * A080910(n). - Michael Somos, Jan 20 2012
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EXAMPLE
| 1 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))} /* Michael Somos, Nov 05 2005 */
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CROSSREFS
| Cf. A089801, A089810
Sequence in context: A002483 A060478 A088806 * A089810 A096562 A096563
Adjacent sequences: A089804 A089805 A089806 * A089808 A089809 A089810
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KEYWORD
| sign
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003
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