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%I #21 Feb 10 2023 17:53:51
%S 0,2,6,36,320,3750,54432,941192,18874368,430467210,11000000000,
%T 311249095212,9659108818944,326173191714734,11905721598812160,
%U 467086816406250000,19599665578316398592,875901453762003632658
%N a(n) = n! * [x^n] W(-x)*(W(-x) + 2)/(W(-x) + 1), where W denotes Lambert's W function.
%D Stephan Wolfram, The Mathematica Book, 4th Edition, Cambridge University Press, section 3.2.10 'Special Functions', page 772, 1999.
%F a(n) = (n+1)*n^(n-1) with a(0) = 0.
%e 2*x + 6*x^2 +36*x^3 + 320*x^4 + 3750*x^5 + 54432*x^6 + 941192*x^7 + ...
%p W := LambertW: egf := -W(-x)*(W(-x) + 2)/(W(-x) + 1):
%p ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # _Peter Luschny_, Feb 10 2023
%t Range[18]!CoefficientList[ Series[ -ProductLog[ -x], {x, 0, 17}], x] (* _Robert G. Wilson v_, Mar 23 2005 *)
%t a[ n_] := If[ n < 0, 0, (n + 1)! SeriesCoefficient[ -ProductLog[-x], {x, 0, n}]] (* _Michael Somos_, Jun 07 2012 *)
%Y Cf. A061250.
%Y Essentially the same as A055541.
%K nonn,easy
%O 0,2
%A Gero Burghardt (gerogoestohollywood(AT)yahoo.de), Jun 05 2001
%E Corrected and extended by _Jason Earls_, Jun 09 2001
%E Name made consistent with offset by _Peter Luschny_, Feb 10 2023
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