A105786 - OEIS

%I #13 Jun 28 2018 04:43:03

%S 0,1,0,3,0,0,16,3,0,0,125,30,0,0,0,1296,330,15,0,0,0,16807,4305,315,0,

%T 0,0,0,262144,66248,5880,105,0,0,0,0,4782969,1183644,115290,3780,0,0,

%U 0,0,0,100000000,24170310,2467080,107100,945,0,0,0,0,0,2357947691,556409535

%N Triangle of the numbers of different forests of m unrooted trees of smallest order 2, i.e., without isolated vertices, on N labeled nodes.

%C 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 Also the Bell transform of A000272(n+1) (with a(0)=0). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016

%F 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-2)Ki) and D = Product_{i=1..N}(Ki!(i!)^Ki).

%e a(8) = 3 because 4 vertices can be partitioned in two trees only in one way: both trees receiving 2 vertices. The unique tree on 2 vertices can be labeled in binomial(4,2) ways and to each one of the binomial(4,2) = 6 possibilities there is just another tree of order 2 in a forest. But since we have 2 trees of the same order, i.e., 2, we must divide binomial(4,2) by 2!.

%p # The function BellMatrix is defined in A264428.

%p # Adds (1,0,0,0, ..) as column 0.

%p BellMatrix(n -> `if`(n=0,0,(n+1)^(n-1)), 9); # _Peter Luschny_, Jan 27 2016

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, If[n == 0, 0, (n+1)^(n-1)]], rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%Y Cf. A033185, A105599.

%K nonn,tabl

%O 1,4

%A _Washington Bomfim_, Apr 21 2005