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A356000 Expansion of e.g.f. -LambertW((1 - exp(2*x))/2).

%I A356000 #23 Feb 16 2025 08:34:03
%S A356000 0,1,4,25,236,3061,50670,1020881,24245576,663290281,20541118266,
%T A356000 710366714773,27135242829436,1134708855427629,51556563327940390,
%U A356000 2529164265815033241,133229047747850647312,7500633471737652798673,449445732625670948076530
%N A356000 Expansion of e.g.f. -LambertW((1 - exp(2*x))/2).
%H A356000 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A356000 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A356000 a(n) = Sum_{k=1..n} 2^(n-k) * k^(k-1) * Stirling2(n,k).
%F A356000 a(n) ~ 2^(n - 1/2) * sqrt(exp(1) + 2) * n^(n-1) / (exp(n) * (log(exp(1) + 2) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022
%F A356000 E.g.f.: Series_Reversion( (log(1 + 2 * x * exp(-x)))/2 ). - _Seiichi Manyama_, Sep 11 2024
%t A356000 With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[2*x])/2], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)
%o A356000 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw((1-exp(2*x))/2))))
%o A356000 (PARI) a(n) = sum(k=1, n, 2^(n-k)*k^(k-1)*stirling(n, k, 2));
%Y A356000 Cf. A048802, A356001.
%Y A356000 Cf. A357335.
%K A356000 nonn
%O A356000 0,3
%A A356000 _Seiichi Manyama_, Sep 24 2022