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A308366 Expansion of Sum_{k>=1} (-1)^(k+1)*k*x^k/(1 - k*x^k). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..45.

FORMULA

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.

MATHEMATICA

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}]

CROSSREFS

Cf. A002129, A055225, A076717.

Sequence in context: A200021 A112518 A228715 * A056849 A116081 A275162

Adjacent sequences:  A308363 A308364 A308365 * A308367 A308368 A308369

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, May 22 2019

STATUS

approved

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Last modified June 17 05:18 EDT 2021. Contains 345080 sequences. (Running on oeis4.)