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

%I A303911 #10 May 21 2018 15:16:34
%S A303911 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 A303911 7,33,103,217,304,262,115,1,8,44,162,423,764,945,727,286,1,9,56,241,
%U A303911 743,1658,2643,2916,2033,719,1,10,70,341,1221,3224,6319,8996,8984,5714,1842,1,11,85,466,1893
%N A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.
%C A303911 Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes.
%H A303911 Andrew Howroyd, <a href="/A303911/b303911.txt">Table of n, a(n) for n = 1..1275</a>
%H A303911 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) 141-162, W(x,y) equation (9a)
%e A303911 The triangle starts
%e A303911 1 ;
%e A303911 1  1 ;
%e A303911 1  2  2 ;
%e A303911 1  3  5   4 ;
%e A303911 1  4 10  13    9 ;
%e A303911 1  5 16  31   35    20 ;
%e A303911 1  6 24  60   98    95    48 ;
%e A303911 1  7 33 103  217   304   262   115 ;
%e A303911 The first column (for a single node n=1) is 1, because all the weight is on that node.
%o A303911 (PARI)
%o A303911 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 A303911 seq(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
%o A303911 {my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ _Andrew Howroyd_, May 19 2018
%Y A303911 Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted).
%K A303911 nonn,tabl
%O A303911 1,5
%A A303911 _R. J. Mathar_, May 02 2018