cp's OEIS Frontend

A357322 Expansion of e.g.f. -LambertW(log(1 - 3*x)/3).

%I A357322 #20 Feb 16 2025 08:34:04
%S A357322 0,1,5,45,586,10024,213084,5428072,161475320,5501761488,211466328400,
%T A357322 9057714349672,428022643010544,22127292215218072,1242503403120434168,
%U A357322 75319473068729478360,4902798528238919060224,341102498012848479889408
%N A357322 Expansion of e.g.f. -LambertW(log(1 - 3*x)/3).
%H A357322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A357322 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A357322 a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * |Stirling1(n,k)|.
%F A357322 a(n) ~ 3^(n - 1/2) * n^(n-1) / ((-1 + exp(3*exp(-1)))^(n - 1/2) * exp(n - 1/2 - 3*n*exp(-1))). - _Vaclav Kotesovec_, Oct 04 2022
%F A357322 E.g.f.: Series_Reversion( (1 - exp(-3 * x * exp(-x)))/3 ). - _Seiichi Manyama_, Sep 11 2024
%t A357322 With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 3*x]/3], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)
%o A357322 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-3*x)/3))))
%o A357322 (PARI) a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));
%Y A357322 Cf. A052807, A357321.
%Y A357322 Cf. A357334.
%K A357322 nonn
%O A357322 0,3
%A A357322 _Seiichi Manyama_, Sep 24 2022