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

2,2

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).

Links

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

Formula

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

Example

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,
...

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 *)

Crossrefs

Cf. A000081, A033185, A174145, A181360.

Sequence in context: A057551 A019823 A338789 * A182903 A169840 A321461

Adjacent sequences: A174132 A174133 A174134 * A174136 A174137 A174138

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 26 2010

Status

approved