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A105820 Triangle giving the numbers of different forests of m trees of smallest order 2, i.e., without isolated vertices.

%I A105820 #19 Feb 16 2025 08:32:57
%S A105820 0,1,0,1,0,0,2,1,0,0,3,1,0,0,0,6,3,1,0,0,0,11,5,1,0,0,0,0,23,12,3,1,0,
%T A105820 0,0,0,47,23,6,1,0,0,0,0,0,106,52,14,3,1,0,0,0,0,0,235,110,29,6,1,0,0,
%U A105820 0,0,0,0,551,253,68,15,3,1,0,0,0,0,0,0,1301,570,148,31,6,1,0,0,0,0,0,0,0
%N A105820 Triangle giving the numbers of different forests of m trees of smallest order 2, i.e., without isolated vertices.
%C A105820 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.
%H A105820 Alois P. Heinz, <a href="/A105820/b105820.txt">Rows n = 1..141, flattened</a>
%H A105820 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Forest.html">Forest</a>
%F A105820 a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and no part equal to 1, of Product_{i=1..N} binomial(A000055(i)+Ki-1, Ki).
%F A105820 G.f.: 1/Product_{i>=2}(1 - x*y^i)^A000055(i). - _Vladeta Jovovic_, Apr 27 2005
%e A105820 a(12) = 1 because 5 nodes can be partitioned into two trees only in one way: one tree gets 3 nodes and the other tree gets 2. Since A000055(3) = A000055(2) = 1, there is only one forest. (The forests of order less than or equal to 5 are depicted in the Weisstein link.)
%Y A105820 Cf. A033185, A095133, A105786, A105821.
%K A105820 nonn,tabl
%O A105820 1,7
%A A105820 _Washington Bomfim_, Apr 25 2005