This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305819 #21 Feb 16 2025 08:33:54
%S A305819 1,1,3,17,132,1334,16442,239994,4041776,77183328,1647541632,
%T A305819 38877352392,1004869488048,28234217634024,856830099396840,
%U A305819 27930093941047464,973269467390922240,36104568839480990400,1420556927968241858880,59088101641333114906944,2590680379402887359111424
%N A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).
%C A305819 Inverse Stirling transform of A000312.
%H A305819 Seiichi Manyama, <a href="/A305819/b305819.txt">Table of n, a(n) for n = 0..397</a>
%H A305819 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305819 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A305819 a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.
%F A305819 a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - _Vaclav Kotesovec_, Aug 18 2018
%p A305819 a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # _Paolo P. Lava_, Mar 26 2019
%t A305819 nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
%t A305819 Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]
%o A305819 (PARI) a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ _Seiichi Manyama_, Feb 05 2022
%o A305819 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ _Seiichi Manyama_, Feb 05 2022
%Y A305819 Cf. A000312, A052807, A277489, A282190, A305981.
%K A305819 nonn
%O A305819 0,3
%A A305819 _Ilya Gutkovskiy_, Aug 18 2018