%I
%S 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,
%T 7,33,103,217,304,262,115,1,8,44,162,423,764,945,727,286,1,9,56,241,
%U 743,1658,2643,2916,2033,719,1,10,70,341,1221,3224,6319,8996,8984,5714,1842,1,11,85,466,1893
%N Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.
%C Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes.
%H Andrew Howroyd, <a href="/A303911/b303911.txt">Table of n, a(n) for n = 1..1275</a>
%H F. Harary, G. Prins, <a href="http://dx.doi.org/10.1007/BF02559543">The number of homeomorphically irreducible trees and other species</a>, Acta Math. 101 (1959) 141162, W(x,y) equation (9a)
%e The triangle starts
%e 1 ;
%e 1 1 ;
%e 1 2 2 ;
%e 1 3 5 4 ;
%e 1 4 10 13 9 ;
%e 1 5 16 31 35 20 ;
%e 1 6 24 60 98 95 48 ;
%e 1 7 33 103 217 304 262 115 ;
%e The first column (for a single node n=1) is 1, because all the weight is on that node.
%o (PARI)
%o 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)}
%o seq(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
%o {my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ _Andrew Howroyd_, May 19 2018
%Y Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted).
%K nonn,tabl
%O 1,5
%A _R. J. Mathar_, May 02 2018
