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A133693
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Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
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0
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1, -1, 2, -1, 0, -2, 0, -1, 3, 0, 2, -2, 0, 0, 0, -1, 2, -3, 2, 0, 0, -2, 0, -2, 1, 0, 4, 0, 0, 0, 0, -1, 4, -2, 0, -3, 0, -2, 0, 0, 2, 0, 2, -2, 0, 0, 0, -2, 1, -1, 4, 0, 0, -4, 0, 0, 4, 0, 2, 0, 0, 0, 0, -1, 0, -4, 2, -2, 0, 0, 0, -3, 2, 0, 2, -2, 0, 0, 0, 0, 5, -2, 2, 0, 0, -2, 0, -2, 2, 0, 0, 0, 0, 0, 0, -2, 2, -1, 6, -1, 0, -4, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n nonzero, a(n) is nonzero if and only if n is in A002479.
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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 (1 - eta(q)^2 * eta(q^4)^5 / ( eta(q^2)^3 * eta(q^8)^2 )) / 2 in powers of q.
Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), a(p^e) = e + 1 if p == 1, 3 (mod 8).
a(8*n + 5) = a(8*n + 7) = 0. A133692(n) = -2 * a(n) unless n=0. -(-1)^n * A002325(n) = a(n). A113411(n) = a(2*n + 1).
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EXAMPLE
| q - q^2 + 2*q^3 - q^4 - 2*q^6 - q^8 + 3*q^9 + 2*q^11 - 2*q^12 - q^16 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -8, d)))}
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CROSSREFS
| Cf. A002325, A002479, A113411, A133692.
Sequence in context: A036577 A002325 A129134 * A065675 A194313 A127476
Adjacent sequences: A133690 A133691 A133692 * A133694 A133695 A133696
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Sep 20 2007
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