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A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

%I A282190 #27 Feb 16 2025 08:33:40
%S A282190 1,1,5,40,447,6421,112726,2338799,55990213,1519122598,46066158817,
%T A282190 1543974969769,56677405835276,2261488166321697,97455090037460785,
%U A282190 4510770674565054000,223183550978156866507,11755122645815049275521,656670295411196201190366,38779502115371642484125915,2413908564514961126280655257
%N A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.
%C A282190 Stirling transform of A000312.
%H A282190 G. C. Greubel, <a href="/A282190/b282190.txt">Table of n, a(n) for n = 0..375</a>
%H A282190 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A282190 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A282190 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A282190 a(0) = 1, a(n) = Sum_{k=1..n} Stirling2(n,k)*k^k.
%F A282190 a(n) ~ n^n / (sqrt(1+exp(1)) * (log(1+exp(-1)))^(n+1/2) * exp(n)). - _Vaclav Kotesovec_, Feb 17 2017
%e A282190 E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...
%p A282190 b:= proc(n, m) option remember;
%p A282190      `if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
%p A282190     end:
%p A282190 a:= n-> b(n, 0):
%p A282190 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 03 2021
%t A282190 Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
%t A282190 Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]
%o A282190 (PARI) x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ _G. C. Greubel_, Nov 12 2017
%Y A282190 Cf. A000312, A000670, A038052, A048802, A052880, A308490, A308491.
%K A282190 nonn,nice
%O A282190 0,3
%A A282190 _Ilya Gutkovskiy_, Feb 08 2017