A061302 a(n) = n! * [x^n] W(-x)*(W(-x) + 2)/(W(-x) + 1), where W denotes Lambert's W function.

0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658

Offset: 0

Author

Gero Burghardt (gerogoestohollywood(AT)yahoo.de), Jun 05 2001

Keywords

• nonn
• easy

Examples

			2*x + 6*x^2 +36*x^3 + 320*x^4 + 3750*x^5 + 54432*x^6 + 941192*x^7 + ...
		

References

• Stephan Wolfram, The Mathematica Book, 4th Edition, Cambridge University Press, section 3.2.10 'Special Functions', page 772, 1999.

Crossrefs

Cf. A061250.

Essentially the same as A055541.

Programs

Maple

W := LambertW: egf := -W(-x)*(W(-x) + 2)/(W(-x) + 1):
ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Feb 10 2023

Mathematica

Range[18]!CoefficientList[ Series[ -ProductLog[ -x], {x, 0, 17}], x] (* Robert G. Wilson v, Mar 23 2005 *)
a[ n_] := If[ n < 0, 0, (n + 1)! SeriesCoefficient[ -ProductLog[-x], {x, 0, n}]] (* Michael Somos, Jun 07 2012 *)

Formula

a(n) = (n+1)*n^(n-1) with a(0) = 0.

Extensions

Corrected and extended by Jason Earls, Jun 09 2001

Name made consistent with offset by Peter Luschny, Feb 10 2023