A384937 Number for rooted ordered trees with edge weights summing to n, where edge weights are all greater than zero, and the sequences of edge weights in all downward paths are weakly increasing.

1, 1, 3, 9, 30, 103, 372, 1379, 5248, 20356, 80252, 320581, 1295018, 5280967, 21711163, 89890559, 374478935, 1568585095, 6602283315, 27910296899, 118448905668, 504466997897, 2155412350793, 9236401247438, 39686616306747, 170946789568804, 738024717474360

Offset: 0

Author

John Tyler Rascoe, Jun 12 2025

Keywords

• nonn

Examples

			The following tree with sum of edge weights 13 contains downward paths of edge weights (1), (2,3,4), and (2,3,3) all of which are weakly increasing. So this tree is counted under a(13) = 5280967.
           o
        2 / \ 1
         o   o
      3 / 	
       o
    4 / \ 3	
     o   o
		

Crossrefs

Cf. A000108, A000169, A000957, A001764, A036765, A384747, A384748.

Programs

PARI

w(j,k,N) = {if(k>N,1, 1/(1 - sum(i=j,N, x^i * w(i,k+1,N-i+1))))}
Ax(N) = {Vec(w(1,1,N)+ O('x^(N+1)))}
Ax(10)

Formula

G.f.: G_1(x) where G_k(x) = 1/(1 - Sum_{i>=k} x^i * G_i(x)).