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 A358738 #15 Feb 18 2023 03:12:49
%S A358738 1,1,3,15,103,893,9341,114355,1603155,25318137,444689497,8597568671,
%T A358738 181430298479,4149361409077,102229328244837,2699254206069387,
%U A358738 76038064580742091,2276259442660623857,72160287650141753777,2414950992007231422007
%N A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.
%F A358738 a(n) = Sum_{k=1..n} k! * k^(n-k) * binomial(n-1,k-1) for n > 0.
%F A358738 a(n) ~ n! / ((1 + LambertW(1))^2 * LambertW(1)^n). - _Vaclav Kotesovec_, Feb 18 2023
%t A358738 nmax = 20; CoefficientList[Series[Sum[k! * (x/(1 - k*x))^k, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 18 2023 *)
%o A358738 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1-k*x))^k))
%o A358738 (PARI) a(n) = if(n==0, 1, sum(k=1, n, k!*k^(n-k)*binomial(n-1, k-1)));
%Y A358738 Cf. A001339, A006153, A080108.
%Y A358738 Cf. A358740, A358741.
%K A358738 nonn
%O A358738 0,3
%A A358738 _Seiichi Manyama_, Nov 29 2022