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 A277509 #22 Feb 16 2025 08:33:36
%S A277509 1,0,4,15,196,2145,33786,587041,12080888,278692497,7213075030,
%T A277509 205967845281,6444486304884,219096784628761,8044651840755362,
%U A277509 317224112769528945,13371158269397088496,599930571306586259745,28547657791777984900014,1436014157616531876023713
%N A277509 Expansion of e.g.f. 1/((1+LambertW(-x))*(1+x)).
%H A277509 G. C. Greubel, <a href="/A277509/b277509.txt">Table of n, a(n) for n = 0..385</a>
%H A277509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A277509 For n > 0, a(n) = (-1)^n*n!+Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^k * (n-k)!.
%F A277509 a(n) ~ n^n / (1+exp(-1)).
%F A277509 a(0) = 1; a(n) = -n*a(n-1) + n^n. - _Seiichi Manyama_, May 01 2023
%t A277509 CoefficientList[Series[1/(1+LambertW[-x])/(1+x), {x, 0, 20}], x] * Range[0, 20]!
%t A277509 Flatten[{1, Table[(-1)^n*n! + Sum[(-1)^(n-k) * Binomial[n, k] * k^k * (n-k)!, {k, 1, n}], {n, 1, 20}]}]
%o A277509 (PARI) my(x='x+O('x^50)); Vec(serlaplace(1/((1 + lambertw(-x))*(1+x)))) \\ _G. C. Greubel_, Nov 12 2017
%Y A277509 Cf. A000312, A277506, A277508.
%K A277509 nonn,easy
%O A277509 0,3
%A A277509 _Vaclav Kotesovec_, Oct 18 2016