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 A319392 #14 May 07 2020 22:35:45
%S A319392 1,0,5,116,4785,307024,28435285,3598112580,596971515329,
%T A319392 125802906617600,32834740225688901,10399056510149276980,
%U A319392 3929349957207906673585,1746371472945523953503376,901944505258819679842017365,535692457387043907059336566724,362573376628272441934460817960705
%N A319392 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*k!*n^k.
%H A319392 Alois P. Heinz, <a href="/A319392/b319392.txt">Table of n, a(n) for n = 0..232</a>
%F A319392 a(n) = n! * [x^n] exp(-x)/(1 - n*x).
%F A319392 a(n) = exp(-1/n)*n^n*Gamma(n+1,-1/n) for n > 0, where Gamma(a,x) is the incomplete gamma function.
%F A319392 a(n) ~ n! * n^n. - _Vaclav Kotesovec_, Jun 09 2019
%p A319392 b:= proc(n, k) option remember;
%p A319392 `if`(n=0, 1, k*n*b(n-1, k)+(-1)^n)
%p A319392 end:
%p A319392 a:= n-> b(n$2):
%p A319392 seq(a(n), n=0..17); # _Alois P. Heinz_, May 07 2020
%t A319392 Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] k! n^k, {k, 0, n}], {n, 16}]]
%t A319392 Table[n! SeriesCoefficient[Exp[-x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
%t A319392 Table[(-1)^n HypergeometricPFQ[{1, -n}, {}, n], {n, 0, 16}]
%o A319392 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*k!*n^k); \\ _Michel Marcus_, Sep 18 2018
%Y A319392 Main diagonal of A320032.
%Y A319392 Cf. A000166, A000180, A000354, A001907, A001908, A277452.
%K A319392 nonn
%O A319392 0,3
%A A319392 _Ilya Gutkovskiy_, Sep 18 2018