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 A290158 #34 May 06 2023 09:00:23
%S A290158 1,0,4,-9,208,-1525,33516,-463099,11293248,-231839577,6517863100,
%T A290158 -175791146311,5723314711632,-189288946716181,7083626583237036,
%U A290158 -275649085963046475,11724766124450058496,-522717581675749841713,24981438186138642481404
%N A290158 a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).
%C A290158 The n-th term of the n-th inverse binomial transform of A000312.
%H A290158 G. C. Greubel, <a href="/A290158/b290158.txt">Table of n, a(n) for n = 0..385</a>
%H A290158 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A290158 a(n) ~ (-1)^n * n^n / (1 + LambertW(1)). - _Vaclav Kotesovec_, Oct 06 2017
%F A290158 From _Seiichi Manyama_, May 05 2023: (Start)
%F A290158 a(n) = (-1)^n * n! * [x^n] exp(n * x * (exp(x) - 1)).
%F A290158 a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} n^k * Stirling2(n-k,k)/(n-k)!.
%F A290158 a(n) = [x^n] Sum_{k>=0} (k*x)^k / (1 + n*x)^(k+1).
%F A290158 a(n) = Sum_{k=0..n} (-n)^(n-k) * k^k * binomial(n,k). (End)
%t A290158 Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]
%o A290158 (PARI) a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ _Seiichi Manyama_, May 05 2023
%Y A290158 Cf. A000312, A069856, A290840, A318615, A356806.
%K A290158 sign
%O A290158 0,3
%A A290158 _Ilya Gutkovskiy_, Oct 06 2017