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Search: id:A133693
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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
1
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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; listen)
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OFFSET
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1,3
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FORMULA
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Expansion of (1 - eta(q)^2 * eta(q^4)^5 / ( eta(q^2)^3 * eta(q^8)^2 )) / 2 in powers of q.
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.
Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].
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EXAMPLE
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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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PROGRAM
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -8, d)))}
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CROSSREFS
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Cf. A133692(n) = -2 * a(n) unless n=0. -(-1)^n * A002325(n) = a(n). A113411(n) = a(2*n+1).
Adjacent sequences: A133690 A133691 A133692 this_sequence A133694 A133695 A133696
Sequence in context: A035158 A002325 A129134 this_sequence A065675 A127476 A140397
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Sep 20 2007
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