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A030202
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Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.
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1
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1, -1, -1, 0, 0, 0, 1, 2, 0, 0, -2, 1, -1, 0, 0, -2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 2, 0, 0, -2, 1, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, -2, 0, 0, 2, -1, -2, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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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
| Bruce Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see page 44.
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
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FORMULA
| Expansion of f(-q, -q^4) * f(-q^2, -q^3) in powers of q where f() is the Ramanujan two variable theta function.
Expansion of q^(-1) * (phi(q) * phi(q^20) - phi(q^4) * phi(q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function.
Euler transform of period 5 sequence [ -1, -1, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80*t)) = 80^(1/2)*(t/i)*f(t) where q = exp(2*Pi*i*t).
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (i^n +(-i)^n)/2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*y) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2. - Michael Somos, Sep 04 2007
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(5*k)).
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EXAMPLE
| 1 - x - x^2 + x^6 + 2*x^7 - 2*x^10 + x^11 - x^12 - 2*x^15 + x^20 + ...
q - q^5 - q^9 + q^25 + 2*q^29 - 2*q^41 + q^45 - q^49 - 2*q^61 + q^81 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ q, q] QPochhammer[ q^5, q^5], {q, 0, n}] (* Michael Somos, Aug 08 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/5, q^2] EllipticTheta[ 1, 2 Pi/5, q^2] / Sqrt[5], {q, 0, 4 n + 1}] // FullSimplify (* Michael Somos, Aug 08 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) * eta(x + x * O(x^n)), n))} /* Michael Somos, Sep 04 2007 */
(PARI) {a(n) = local(A, p, e, x, y); if( n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p==5, (-1)^e, if( p%20>10, !(e%2), if( p%4==3, kronecker( -4, e+1), for( y=1, sqrtint(p\5), if( issquare(p - 5*y^2), x=y; break)); (-1)^(e*x) *(e+1))))))))} /* Michael Somos, Sep 04 2007 */
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CROSSREFS
| Sequence in context: A156996 A029304 * A159818 A081827 A100286 A029303
Adjacent sequences: A030199 A030200 A030201 * A030203 A030204 A030205
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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