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A030203
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Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.
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0
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1, -1, -1, -1, 1, 2, -1, 2, 0, 0, -1, -1, -1, -1, 0, 1, -1, -1, 2, 0, 1, 2, 1, -1, 0, -1, 2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 2, 2, 0, -1, 1, 0, 1, 0, -1, 2, 2, 1, 0, -2, 2, -1, 0, -1, -1, -1, 1, -1, 0, 0, -1, -1, -1, 0, 0, 2, -2, -1, 0, -1, 1, 2, 2, 0, 0, 2, -1, 1, -1, 1, -1, 0, 2, 0, 1, 0, -1, 1, 0, -1, -1, 1, -2, 0, -1, -1, 0, -2, 0, 2, -1, -1, -1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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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
| M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
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FORMULA
| Expansion of f(-x) * f(-x^3) where f(-x) = f(-x, -x^2) is a Ramanujan theta function. - Michael Somos, Jul 27 2006
Expansion of q^(-1/6) * sqrt(b(q) * c(q)/3) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Euler transform of period 3 sequence [ -1, -1, -2, ...]. - Michael Somos, Jul 27 2006
Given g.f. A(x), then B(x) = (x * A(x^6))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 - u^2*w - 4*u*w^2 . - Michael Somos, Jul 27 2006
a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p = x^2 + 27*y^2, b(p^e) = [1, -1, 0] depending on e (mod 3) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108^(1/5) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Jan 22 2012
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)).
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EXAMPLE
| 1 - x - x^2 - x^3 + x^4 + 2*x^5 - x^6 + 2*x^7 - x^10 - x^11 - x^12 - x^13 + ...
q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - q^67 - ...
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PROG
| (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p<5, 0, if( p%6==5, (1 + (-1)^e)/2, if( (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1))))))} /* Michael Somos, Jul 27 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A), n))} /* Michael Somos, Jul 27 2006 */
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CROSSREFS
| Sequence in context: A035186 A035194 A161491 * A101664 A091952 A108803
Adjacent sequences: A030200 A030201 A030202 * A030204 A030205 A030206
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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