A174135 Irregular triangle read by rows: T(n,k), n >= 2, 1 <= k <= n/2, = number of rooted forests with n nodes and k trees, with at least two nodes in each tree.

1, 2, 4, 1, 9, 2, 20, 7, 1, 48, 17, 2, 115, 48, 7, 1, 286, 124, 21, 2, 719, 336, 60, 7, 1, 1842, 888, 171, 21, 2, 4766, 2393, 488, 65, 7, 1, 12486, 6419, 1372, 187, 21, 2, 32973, 17376, 3862, 554, 65, 7, 1, 87811, 47097, 10846, 1600, 193, 21, 2, 235381, 128365, 30429, 4644, 574, 65, 7, 1, 634847, 350837, 85365, 13362, 1685, 193, 21, 2

Offset: 2

Author

N. J. A. Sloane, Nov 26 2010

Keywords

• nonn
• tabf

Comments

In other words, components consisting of just a root node are forbidden. If this condition is removed, we get A033185.

First column is a version of A000081. Row sums give A174145.

Diagonal sums give A181360 (e.g., 9+7+2+1 = 19).

Examples

			Triangle begins:
1,
2,
4, 1,
9, 2,
20, 7, 1,
48, 17, 2,
115, 48, 7, 1,
286, 124, 21, 2,
719, 336, 60, 7, 1,
1842, 888, 171, 21, 2,
4766, 2393, 488, 65, 7, 1,
12486, 6419, 1372, 187, 21, 2,
32973, 17376, 3862, 554, 65, 7, 1,
87811, 47097, 10846, 1600, 193, 21, 2,
235381, 128365, 30429, 4644, 574, 65, 7, 1,
634847, 350837, 85365, 13362, 1685, 193, 21, 2,
1721159, 962731, 239566, 38459, 4948, 581, 65, 7, 1,
...
		

Links

• Alois P. Heinz, Rows n = 2..200, flattened

Crossrefs

Cf. A000081, A033185, A174145, A181360.

Programs

Maple

with(numtheory):
t:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(p<1 or i<2, 0, add(b(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p) ))))
    end:
T:= (n, k)-> b(n, n, k):
seq(seq(T(n, k), k=1..iquo(n, 2)), n=2..18);  # Alois P. Heinz, May 17 2013

Mathematica

t[n_] := t[n] = Module[{d, j}, If[n <= 1, n, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]]; b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[p < 1 || i < 2, 0, Sum[b[n-i*j, i-1, p-j]* Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; T[n_, k_] := b[n, n, k]; Table[Table[T[n, k], {k, 1, Quotient[n, 2]}], {n, 2, 18}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Formula

G.f.: 1/Product((1-x*y^i)^A000081(i), i=2..infinity).