A308367 Expansion of Sum_{k>=1} x^k/(1 + k*x^k).

1, 0, 2, -2, 2, 1, 2, -12, 11, 11, 2, -49, 2, 57, 108, -200, 2, 40, 2, -391, 780, 1013, 2, -5423, 627, 4083, 6644, -4453, 2, -5043, 2, -49680, 59172, 65519, 18028, -251062, 2, 262125, 531612, -861481, 2, -515723, 2, -1049929, 5180382, 4194281, 2, -27246019, 117651

Offset: 1

Author

Ilya Gutkovskiy, May 22 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A076717, A087909, A300786.

Programs

Mathematica

nmax = 49; CoefficientList[Series[Sum[x^k /(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 49; CoefficientList[Series[Log[Product[(1 + k x^k)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-d)^(n/d - 1), {d, Divisors[n]}], {n, 1, 49}]

PARI

a(n) = sumdiv(n, d, (-d)^(n/d-1)); \\ Michel Marcus, Mar 22 2021

Formula

L.g.f.: log(Product_{k>=1} (1 + k*x^k)^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.

a(n) = Sum_{d|n} (-d)^(n/d-1).

a(n) = 2 if n is odd prime.