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A357321 Expansion of e.g.f. -LambertW(log(1 - 2*x)/2).

%I A357321 #19 Feb 16 2025 08:34:04
%S A357321 0,1,4,29,308,4349,77094,1650893,41532280,1201865049,39351776970,
%T A357321 1438731784137,58107225611412,2569486856423733,123475320944016846,
%U A357321 6407225728624769925,357061085760608504304,21268522319028809507889,1348496822257863921774738
%N A357321 Expansion of e.g.f. -LambertW(log(1 - 2*x)/2).
%H A357321 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A357321 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A357321 a(n) = Sum_{k=1..n} 2^(n-k) * k^(k-1) * |Stirling1(n,k)|.
%F A357321 a(n) ~ 2^(n - 1/2) * n^(n-1) / ((-1 + exp(2*exp(-1)))^(n - 1/2) * exp(n - 2*n*exp(-1) - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022
%F A357321 E.g.f.: Series_Reversion( (1 - exp(-2 * x * exp(-x)))/2 ). - _Seiichi Manyama_, Sep 11 2024
%t A357321 With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 2*x]/2], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)
%o A357321 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-2*x)/2))))
%o A357321 (PARI) a(n) = sum(k=1, n, 2^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));
%Y A357321 Cf. A052807, A357322.
%Y A357321 Cf. A357333.
%K A357321 nonn
%O A357321 0,3
%A A357321 _Seiichi Manyama_, Sep 24 2022