A106234 Triangle of the numbers of different forests with one or more isolated vertices. Those forests of rooted trees, have order N and m trees.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 9, 6, 3, 1, 1, 0, 20, 16, 7, 3, 1, 1, 0, 48, 37, 18, 7, 3, 1, 1, 0, 115, 96, 44, 19, 7, 3, 1, 1, 0, 286, 239, 117, 46, 19, 7, 3, 1, 1, 0, 719, 622, 299, 124, 47, 19, 7, 3, 1, 1, 0, 1842, 1607, 793, 320, 126, 47, 19, 7, 3, 1, 1

Offset: 1

Author

Washington Bomfim, Apr 26 2005

Keywords

• nonn
• tabl

Comments

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.

Examples

			a(13) = 3 because 5 vertices can be partitioned in 3 trees in two ways: (1) one tree gets 3 nodes and the others get 1 each. (2) two trees get 2 nodes each and the other gets 1. Case (1) corresponds to 2 forests since A000081(3) = 2. Case (2) corresponds to one forest since A000081(2) = 1.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   4,   3,   1,  1;
  0,   9,   6,   3,  1,  1;
  0,  20,  16,   7,  3,  1, 1;
  0,  48,  37,  18,  7,  3, 1, 1;
  0, 115,  96,  44, 19,  7, 3, 1, 1;
  0, 286, 239, 117, 46, 19, 7, 3, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A033185, A105820.

Row sums give A000081.

Programs

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n<=1, n, (add(add(
      d*g(d), d=divisors(j))*g(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 0, `if`(i=1,
      x^n, expand(add(x^j*b(n-i*j, i-1)*
      binomial(g(i)+j-1,j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Jun 25 2014

Mathematica

g[n_] := g[n] = If[n <= 1, n, (Sum[Sum[d*g[d], {d, Divisors[j]}]*g[n-j], {j, 1, n-1}])/(n-1)]; b[n_, i_] := b[n, i] = If[n == 0, 0, If[i == 1, x^n, Expand[ Sum[ x^j*b[n-i*j, i-1]*Binomial[g[i]+j-1, j], {j, 0,  n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Formula

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(A000081(i)+Ki-1,Ki).