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 A105819 #19 May 03 2023 17:13:22
%S A105819 0,2,0,9,0,0,64,12,0,0,625,180,0,0,0,7776,2730,120,0,0,0,117649,46410,
%T A105819 3780,0,0,0,0,2097152,893816,99120,1680,0,0,0,0,43046721,19389384,
%U A105819 2600640,90720,0,0,0,0,0,1000000000,469532790,71734320,3654000,30240,0
%N A105819 Triangle of the numbers of different forests of m rooted trees of smallest order 2, i.e., without isolated vertices, on N labeled nodes.
%C A105819 Forests of order N with m components, m > floor(N/2) must contain an isolated vertex since it is impossible to partition N vertices in floor(N/2) + 1 or more trees without giving only one vertex to a tree.
%C A105819 Also the Bell transform of A055860. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%H A105819 Alois P. Heinz, <a href="/A105819/b105819.txt">Rows n = 1..141, flattened</a>
%F A105819 a(n)= 0, if m > floor(N/2) (see comments), or can be calculated by the sum Num/D over the partitions of N: 1K1 + 2K2 + ... + nKN, with exactly m parts and smallest part = 2, where Num = N!*Product_{i=1..N}i^((i-1)Ki) and D = Product_{i=1..N}(Ki!(i!)^Ki).
%F A105819 From _Mélika Tebni_, Apr 23 2023: (Start)
%F A105819 E.g.f. of column k: (-x - LambertW(-x))^k / k!, k > 0.
%F A105819 Sum_{k=1..n} (-1)^(n-k)*T(n+k,k) = n+1.
%F A105819 Sum_{k=1..n} (-1)^(k+1)*T(n,k) = A360193(n), for n > 0.
%F A105819 Sum_{k=1..n} (-1)^(k+1)*T(n+k,k)/(n+k-1) = 1/n, for n > 1.
%F A105819 T(n,k) = Sum_{j=k..n} j!*abs(Stirling1(j-k,k))*A354794(n,j)/(j-k)!. (End)
%e A105819 a(8) = 12 because 4 vertices can be partitioned in two trees only in one way: both trees receiving 2 vertices. Two trees on 2 vertices can be labeled in binomial(4,2) ways and to each one of the 2*binomial(4,2) = 12 possibilities there are more 2 possible trees of order 2 in a forest. But since we have 2 trees of the same order, i.e., 2, we must divide 2*binomial(4,2)*2 by 2!.
%e A105819 Triangle T(n,k) begins:
%e A105819 : 0;
%e A105819 : 2, 0;
%e A105819 : 9, 0, 0;
%e A105819 : 64, 12, 0, 0;
%e A105819 : 625, 180, 0, 0, 0;
%e A105819 : 7776, 2730, 120, 0, 0, 0;
%e A105819 : 117649, 46410, 3780, 0, 0, 0, 0;
%e A105819 : 2097152, 893816, 99120, 1680, 0, 0, 0, 0;
%p A105819 # The function BellMatrix is defined in A264428.
%p A105819 # Adds (1,0,0,0, ..) as column 0.
%p A105819 BellMatrix(n -> `if`(n=0,0,(n+1)^n), 9); # _Peter Luschny_, Jan 27 2016
%p A105819 # second Maple program:
%p A105819 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A105819 binomial(n-1, j-1)*j^(j-1)*x*b(n-j), j=2..n)))
%p A105819 end:
%p A105819 T:= (n, k)-> coeff(b(n), x, k):
%p A105819 seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, Aug 13 2017
%t A105819 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A105819 rows = 12;
%t A105819 B = BellMatrix[Function[n, If[n == 0, 0, (n+1)^n]], rows];
%t A105819 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y A105819 Cf. A033185, A055860, A105599, A354794, A360193.
%Y A105819 Row sums give A105785.
%K A105819 nonn,tabl
%O A105819 1,2
%A A105819 _Washington Bomfim_, Apr 21 2005