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

Author

R. J. Mathar, May 02 2018

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

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)

Crossrefs

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

Programs

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