A358185 Coefficients of x^n/n! in the expansion of (1 - x)*log(1 - x).

0, -1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000

Offset: 0

Author

Wolfdieter Lang, Nov 14 2022

Keywords

• sign
• easy

Comments

The negated sequence gives the compositional inverse of 1 - exp(W(-x)) with the principal branch of Lambert's W function.

See A000312 for the two formulas of Vladimir Kruchinin, Sep 15 2010, and Peter Bala, Dec 09 2011. Here stated the other way around.

Links

• R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], (2018) eq (60).

Crossrefs

Cf. A000142, A103505, A000312, A159333, A104150.

Programs

Mathematica

With[{m = 25}, Range[0, m]! * CoefficientList[Series[(1-x) * Log[1-x], {x, 0, m}], x]] (* Amiram Eldar, Nov 14 2022 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)*log(1-x)), -N) \\ Michel Marcus, Sep 16 2023

Formula

E.g.f.: (1 - x) * log(1 - x).

a(0) = 0, a(1) = -1, a(n) = (n-2)! = A000142(n-2), for n >= 2.