A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 10, 13, 9, 1, 5, 16, 31, 35, 20, 1, 6, 24, 60, 98, 95, 48, 1, 7, 33, 103, 217, 304, 262, 115, 1, 8, 44, 162, 423, 764, 945, 727, 286, 1, 9, 56, 241, 743, 1658, 2643, 2916, 2033, 719, 1, 10, 70, 341, 1221, 3224, 6319, 8996, 8984, 5714, 1842, 1, 11, 85, 466, 1893

Offset

1,5

Comments

Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a)

Example

The triangle starts
1 ;
1  1 ;
1  2  2 ;
1  3  5   4 ;
1  4 10  13    9 ;
1  5 16  31   35    20 ;
1  6 24  60   98    95    48 ;
1  7 33 103  217   304   262   115 ;
The first column (for a single node n=1) is 1, because all the weight is on that node.

Prog

(PARI)
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
{my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ Andrew Howroyd, May 19 2018

Crossrefs

Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted).

Sequence in context: A026300 A099514 A228352 * A205575 A368338 A344583

Adjacent sequences: A303908 A303909 A303910 * A303912 A303913 A303914

Keyword

nonn,tabl

Author

R. J. Mathar, May 02 2018

Status

approved