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 A008827 #55 Apr 01 2023 09:11:35
%S A008827 0,3,13,50,201,875,4138,21145,115973,678568,4213595,27644435,
%T A008827 190899320,1382958543,10480142145,82864869802,682076806157,
%U A008827 5832742205055,51724158235370,474869816156749,4506715738447321,44152005855084344,445958869294805287,4638590332229999351
%N A008827 Number of proper partitions of a set of n labeled elements.
%C A008827 Previous name: Coefficients from fractional iteration of exp(x) - 1.
%C A008827 From _Harry Richman_, Mar 18 2023: (Start)
%C A008827 A "proper partition" of a set is a set partition in which there is more than one part, and there is some part which has more than one element.
%C A008827 a(n) is the number of chains of length 2 from the top element to the bottom element in the partition lattice on n labeled objects.
%C A008827 (End)
%D A008827 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
%H A008827 Vincenzo Librandi, <a href="/A008827/b008827.txt">Table of n, a(n) for n = 2..200</a>
%H A008827 Ivo Rosenberg, <a href="http://dx.doi.org/10.1016/0097-3165(73)90058-7">The number of maximal closed classes in the set of functions over a finite domain</a>, J. Combinatorial Theory Ser. A 14 (1973), 1-7. See Table I (it is not certain that this is the same sequence. - _N. J. A. Sloane_, Jun 25 2015)
%H A008827 Ivo Rosenberg and N. J. A. Sloane, <a href="/A002824/a002824_1.pdf">Correspondence, 1971</a>
%F A008827 a(n) = A000110(n) - 2.
%e A008827 For n = 3 there are a(3) = 3 proper partitions of {1,2,3}, which can be represented {12|3}, {13|2}, {23|1}.
%e A008827 For n = 4 there are a(4) = 13 proper partitions of {1,2,3,4}, which can be represented {123|4}, {124|3}, {134|2}, {234|1}, {12|34}, {13|24}, {14|23}, {12|3|4}, {13|2|4}, {14|2|3}, {23|1|4}, {24|1|3}, {34|1|2}.
%p A008827 seq(combinat[bell](n)-2, n=2..31); # _Zerinvary Lajos_, Sep 29 2006
%t A008827 Table[BellB[n] - 2, {n, 3, 30}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%o A008827 (PARI) a(n)=sum(j=2,n--,(j+1)*stirling(n,j,2)) \\ _Charles R Greathouse IV_, Jul 06 2011
%o A008827 (Magma) [Bell(n) -2: n in [3..30]]; // _G. C. Greubel_, Sep 13 2019
%o A008827 (Sage) [bell_number(n)-2 for n in (3..30)] # _G. C. Greubel_, Sep 13 2019
%o A008827 (GAP) List([3..30], n-> Bell(n)-2); # _G. C. Greubel_, Sep 13 2019
%Y A008827 Cf. A000110, A008826.
%K A008827 nonn
%O A008827 2,2
%A A008827 _N. J. A. Sloane_
%E A008827 More terms from _Vladeta Jovovic_, Jan 02 2004
%E A008827 Name changed and a(2)=0 prepended by _Harry Richman_, Mar 18 2023