A357322 - OEIS

%I #19 Sep 11 2024 10:04:59

%S 0,1,5,45,586,10024,213084,5428072,161475320,5501761488,211466328400,

%T 9057714349672,428022643010544,22127292215218072,1242503403120434168,

%U 75319473068729478360,4902798528238919060224,341102498012848479889408

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * |Stirling1(n,k)|.

%F 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 E.g.f.: Series_Reversion( (1 - exp(-3 * x * exp(-x)))/3 ). - _Seiichi Manyama_, Sep 11 2024

%t With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 3*x]/3], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-3*x)/3))))

%o (PARI) a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));

%Y Cf. A052807, A357321.

%Y Cf. A357334.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 24 2022