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A089802
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Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.
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4
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1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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
| Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(-x) * psi(x^3) in powers of x. - Michael Somos, Dec 23 2011
Expansion of f(-x, -x^5) in powers of x, where f(a, b) = Sum_{k} a^((k^2 + k)/2) * b^((k^2 - k)/2) is Ramanujan's two-variable theta function.
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A089812. - Michael Somos, Dec 23 2011
Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, -1, ...]. - Michael Somos, Apr 12 2005
|a(n)| is the characteristic function of A001082. - Michael Somos, Oct 31 2005
G.f.: Sum_{k} (-1)^k * x^((3k^2 - 2*k) / 2) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)) . - Michael Somos, Oct 31 2005
A002448(3*n + 1) = -2 * a(n). - Michael Somos, Jul 07 2006
a(n) = (-1)^n * A089801(n).
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EXAMPLE
| 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^56 - x^65 - x^85 + ...
q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + q^121 + q^169 - q^196 + ...
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PROG
| (PARI) {a(n) = (-1)^n * issquare(3*n + 1) /* Michael Somos, Apr 12 2005 */
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CROSSREFS
| Cf. A001082, A002448, A089801, A089812.
Sequence in context: * A143064 A185124 A185125 A163811 A163817 A151667
Adjacent sequences: A089799 A089800 A089801 * A089803 A089804 A089805
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KEYWORD
| sign
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003
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EXTENSIONS
| Corrected by N. J. A. Sloane (njas(AT)research.att.com), Nov 05 2005
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