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 A348311 #6 Oct 21 2021 01:25:25
%S A348311 0,1,2,3,20,85,534,3703,29672,266985,2669930,29369131,352429692,
%T A348311 4581585853,64142202110,962133031455,15394128503504,261700184559313,
%U A348311 4710603322067922,89501463119290195,1790029262385804260,37590614510101889061,826993519222241559782
%N A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.
%F A348311 E.g.f.: x * (1 + x) * exp(-x) / (1 - x).
%F A348311 a(0) = 0; a(n) = n * (a(n-1) + (-1)^n * (n-2)).
%F A348311 a(n) = n * (2 * A000166(n-1) + (-1)^n).
%t A348311 Table[n! Sum[(-1)^k (k - 2)/(k - 1)!, {k, 1, n}], {n, 0, 22}]
%t A348311 nmax = 22; CoefficientList[Series[x (1 + x) Exp[-x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%o A348311 (PARI) a(n) = n!*sum(k=1, n, (-1)^k * (k-2) / (k-1)!); \\ _Michel Marcus_, Oct 20 2021
%Y A348311 Cf. A000166, A000240, A030297, A053817, A180189, A212291.
%K A348311 nonn
%O A348311 0,3
%A A348311 _Ilya Gutkovskiy_, Oct 11 2021