cp's OEIS Frontend

A105821 Triangle of the numbers of different forests with one or more isolated vertices. Those forests have order N and m trees.

%I A105821 #22 Feb 16 2025 08:32:57
%S A105821 1,0,1,0,1,1,0,1,1,1,0,2,2,1,1,0,3,3,2,1,1,0,6,6,4,2,1,1,0,11,11,7,4,
%T A105821 2,1,1,0,23,23,14,8,4,2,1,1,0,47,46,29,15,8,4,2,1,1,0,106,99,60,32,16,
%U A105821 8,4,2,1,1,0,235,216,128,66,33,16,8,4,2,1,1,0,551,488,284,143,69,34,16,8,4,2,1,1
%N A105821 Triangle of the numbers of different forests with one or more isolated vertices. Those forests have order N and m trees.
%C A105821 The unique tree with an isolated node has order one. For N > 1 and m > 1 there is at least one partition of N in m parts, with a part equal to 1, so a(n) > 0, when m > 1 and a(n) = 0, when m = 1 and N > 1. A095133(n) = A105821(n) + A105820(n).
%C A105821 a(2*n+1,n+1) = A215930(n) for n>=0. - _Alois P. Heinz_, Jul 10 2013
%H A105821 Alois P. Heinz, <a href="/A105821/b105821.txt">Rows n = 1..141, flattened</a>
%H A105821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Forest.html">Forest</a>
%F A105821 a(n) = sum over the partitions of N: 1K1 + 2K2 + ... + NKN, with exactly m parts and one or more parts equal to 1, of Product_{i=1..N} binomial(A000055(i)+Ki-1, Ki).
%e A105821 a(5,2) = 2 because 5 vertices can be partitioned in two trees only in one way: one tree gets 4 nodes and the other tree gets 1. Since A000055(4) = 2 and A000055(1) = 1, there are 2 forests. The forests of order less than or equal to 5 are depicted in the Weisstein "Forest" link.
%e A105821 1;
%e A105821 0, 1;
%e A105821 0, 1, 1;
%e A105821 0, 1, 1, 1;
%e A105821 0, 2, 2, 1, 1;
%e A105821 0, 3, 3, 2, 1, 1;
%e A105821 0, 6, 6, 4, 2, 1, 1;
%e A105821 0, 11, 11, 7, 4, 2, 1, 1;
%e A105821 0, 23, 23, 14, 8, 4, 2, 1, 1;
%e A105821 0, 47, 46, 29, 15, 8, 4, 2, 1, 1;
%e A105821 0, 106, 99, 60, 32, 16, 8, 4, 2, 1, 1;
%e A105821 0, 235, 216, 128, 66, 33, 16, 8, 4, 2, 1, 1;
%Y A105821 Cf. A095133, A105820, A215930, row-reversed variant of A136605.
%K A105821 nonn,tabl
%O A105821 1,12
%A A105821 _Washington Bomfim_, Apr 25 2005