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Search: id:A137608
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| A137608 |
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Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function. |
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+0
1
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| 1, -1, 1, -1, 0, -1, 2, -1, 1, 0, 0, -1, 2, -2, 0, -1, 0, -1, 2, 0, 2, 0, 0, -1, 1, -2, 1, -2, 0, 0, 2, -1, 0, 0, 0, -1, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, -1, 3, -1, 0, -2, 0, -1, 0, -2, 2, 0, 0, 0, 2, -2, 2, -1, 0, 0, 2, 0, 0, 0, 0, -1, 2, -2, 1, -2, 0, -2, 2, 0, 1, 0, 0, -2, 0, -2, 0, 0, 0, 0, 4, 0, 2, 0, 0, -1, 2, -3, 0, -1, 0, 0, 2, -2, 0
(list; graph; listen)
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OFFSET
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1,7
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FORMULA
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Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless e=0, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(6*n+5) = 0.
G.f.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
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EXAMPLE
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q - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...
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PROGRAM
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(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-12, d)))}
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CROSSREFS
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Cf. -(-1)^n * A035178(n) = a(n). A132973(n) = -3 * a(n) unless n=0.
Sequence in context: A035178 A093829 A113447 this_sequence A078807 A029422 A117452
Adjacent sequences: A137605 A137606 A137607 this_sequence A137609 A137610 A137611
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Jan 29 2008
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