A264234 Numerators of the coefficients in the expansion of 1/W(x) - 1/x where W(x) is the Lambert W function.

1, -1, 2, -9, 32, -625, 324, -117649, 131072, -4782969, 1562500, -25937424601, 35831808, -23298085122481, 110730297608, -4805419921875, 562949953421312, -48661191875666868481, 91507169819844, -104127350297911241532841, 640000000000000000, -865405750887126927009

Offset: 0

Author

Peter Luschny, Nov 09 2015

Keywords

• sign
• frac

Comments

If prefixed by an additional 1, numerators of coefficients of expansion of exp(W(x)). - N. J. A. Sloane, Jan 08 2021

Examples

			Coefficients of expansion of exp(W(x)) are 1, 1, -1/2, 2/3, -9/8, 32/15, -625/144, 324/35, -117649/5760, 131072/2835, -4782969/44800, ... - _N. J. A. Sloane_, Jan 08 2021
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..388
• Wikipedia, Lambert W function

Crossrefs

Denominators in A264235.

Cf. A036505.

Programs

Magma

[(-1)^n * Numerator(n^n/Factorial(n)): n in [0..50]]; // G. C. Greubel, Nov 14 2017

Maple

seq(numer((-1)^n*n^n/n!), n = 0..21);

Mathematica

CoefficientList[Series[1/ProductLog[x] - 1/x, {x, 0, 21}], x] // Numerator

PARI

vector(22, n, n--; (-1)^n*numerator(n^n/n!)) \\ Altug Alkan, Nov 09 2015

Formula

a(n) = (-1)^n*numerator(g(n)) where g(n) = n^n/n!.

a(n) = (-1)^n*denominator(h(n)) where h(n) = Sum_{k=0..n-1}(n!*n^k)/(k!*n^n).