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A308366
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Expansion of Sum_{k>=1} (-1)^(k+1)*k*x^k/(1 - k*x^k).
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1
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1, -1, 4, -7, 6, -4, 8, -39, 37, -16, 12, -94, 14, -92, 384, -591, 18, 65, 20, -1542, 2552, -1948, 24, -3606, 3151, -8048, 20440, -30590, 30, 33326, 32, -135455, 178512, -130816, 94968, -35029, 38, -523964, 1596560, -1749734, 42, 2521186, 44, -8374494, 16364502
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OFFSET
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1,3
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LINKS
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FORMULA
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L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^((-1)^(k+1)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(d+1)*d^(n/d).
a(n) = n + 1 if n is odd prime.
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MATHEMATICA
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nmax = 45; CoefficientList[Series[Sum[(-1)^(k + 1) k x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 45; CoefficientList[Series[-Log[Product[(1 - k x^k)^((-1)^(k + 1)/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-1)^(d + 1) d^(n/d), {d, Divisors[n]}], {n, 1, 45}]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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