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 A332188 #25 Oct 31 2020 02:03:31
%S A332188 0,3,72,1557,36928,986550,29641608,994006209,36887753216,
%T A332188 1502798312547,66730937637400,3209318261685690,166242143849148864,
%U A332188 9229638177763268395,546842961612529341032,34443269219453881669425
%N A332188 a(n) = (1/e^n) * Sum_{j>=2} j^n * n^j / (j-2)!.
%F A332188 a(n) = Sum_{k=0..n+2} n^k*(Stirling2(n+2,k) - Stirling2(n+1,k)). [Thanks to _Andrew Howroyd_ for his example in A338282]
%e A332188 a(3) = 1557 = (1/e^3) * Sum_{j>=2} j^3 * 3^j / factorial(j-2).
%t A332188 a[n_] := Sum[n^k*(StirlingS2[n + 2, k] - StirlingS2[n + 1, k]), {k, 2, n + 2}]; Array[a, 16, 0] (* _Amiram Eldar_, Oct 30 2020 *)
%o A332188 (SageMath) # Increase precision for larger n!
%o A332188 R = RealField(100)
%o A332188 t = 2
%o A332188 sol = [0]*18
%o A332188 for n in range(0, 18):
%o A332188 suma = R(0)
%o A332188 for j in range(t, 1000):
%o A332188 suma += (j^n * n^j) / factorial(j - t)
%o A332188 suma *= exp(-n)
%o A332188 sol[n] = round(suma)
%o A332188 print(sol) # Thanks to _Peter Luschny_ for his example in A338282.
%o A332188 (PARI) a(n) = sum(k=0, n+2, n^k*(stirling(n+2,k,2) - stirling(n+1,k,2))); \\ _Michel Marcus_, Oct 30 2020
%Y A332188 Cf. A005494, A143495, A242817, A299824, A338282.
%K A332188 nonn
%O A332188 0,2
%A A332188 _Pedro Caceres_, Oct 30 2020