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 A354437 #17 May 29 2022 01:57:45
%S A354437 1,1,-1,1,13,-199,2251,-19991,7001,7530193,-330734249,11005284401,
%T A354437 -300961551131,4886902605001,184195977487523,-28517140157423399,
%U A354437 2322376314679777201,-153646291657993064671,8388000381774954552751,-287686436757241322569247
%N A354437 a(n) = n! * Sum_{k=0..n} (-k)^(n-k)/k!.
%F A354437 E.g.f.: Sum_{k>=0} x^k / (k! * (1 + k*x)).
%t A354437 Join[{1}, Table[n!*Sum[ (-k)^(n - k)/k!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 28 2022 *)
%o A354437 (PARI) a(n) = n!*sum(k=0, n, (-k)^(n-k)/k!);
%o A354437 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1+k*x)))))
%o A354437 (Python)
%o A354437 from math import factorial
%o A354437 def A354437(n): return sum(factorial(n)*(-k)**(n-k)//factorial(k) for k in range(n+1)) # _Chai Wah Wu_, May 28 2022
%Y A354437 Cf. A038125, A277509, A354436.
%K A354437 sign
%O A354437 0,5
%A A354437 _Seiichi Manyama_, May 28 2022