OFFSET
1,2
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.
Also the Bell transform of A055860. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
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).
From Mélika Tebni, Apr 23 2023: (Start)
E.g.f. of column k: (-x - LambertW(-x))^k / k!, k > 0.
Sum_{k=1..n} (-1)^(n-k)*T(n+k,k) = n+1.
Sum_{k=1..n} (-1)^(k+1)*T(n,k) = A360193(n), for n > 0.
Sum_{k=1..n} (-1)^(k+1)*T(n+k,k)/(n+k-1) = 1/n, for n > 1.
T(n,k) = Sum_{j=k..n} j!*abs(Stirling1(j-k,k))*A354794(n,j)/(j-k)!. (End)
EXAMPLE
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!.
Triangle T(n,k) begins:
: 0;
: 2, 0;
: 9, 0, 0;
: 64, 12, 0, 0;
: 625, 180, 0, 0, 0;
: 7776, 2730, 120, 0, 0, 0;
: 117649, 46410, 3780, 0, 0, 0, 0;
: 2097152, 893816, 99120, 1680, 0, 0, 0, 0;
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 0, (n+1)^n), 9); # Peter Luschny, Jan 27 2016
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
binomial(n-1, j-1)*j^(j-1)*x*b(n-j), j=2..n)))
end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Aug 13 2017
MATHEMATICA
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[n == 0, 0, (n+1)^n]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Washington Bomfim, Apr 21 2005
STATUS
approved