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 A105820 Triangle giving the numbers of different forests of m trees of smallest order 2, i.e., without isolated vertices. 3
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 1..141, flattened Eric Weisstein's World of Mathematics, Forest FORMULA 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). G.f.: 1/Product_{i>=2}(1 - x*y^i)^A000055(i). - Vladeta Jovovic, Apr 27 2005 EXAMPLE 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.) CROSSREFS Cf. A033185, A095133, A105786, A105821. Sequence in context: A145677 A128229 A132013 * A136263 A105593 A029371 Adjacent sequences: A105817 A105818 A105819 * A105821 A105822 A105823 KEYWORD nonn,tabl AUTHOR Washington Bomfim, Apr 25 2005 STATUS approved

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Last modified September 19 16:10 EDT 2024. Contains 376013 sequences. (Running on oeis4.)