cp's OEIS Frontend

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.

%I A384937 #7 Jun 14 2025 00:18:31
%S A384937 1,1,3,9,30,103,372,1379,5248,20356,80252,320581,1295018,5280967,
%T A384937 21711163,89890559,374478935,1568585095,6602283315,27910296899,
%U A384937 118448905668,504466997897,2155412350793,9236401247438,39686616306747,170946789568804,738024717474360
%N 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.
%F A384937 G.f.: G_1(x) where G_k(x) = 1/(1 - Sum_{i>=k} x^i * G_i(x)).
%e A384937 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.
%e A384937            o
%e A384937         2 / \ 1
%e A384937          o   o
%e A384937       3 / 	
%e A384937        o
%e A384937     4 / \ 3	
%e A384937      o   o
%o A384937 (PARI)
%o A384937 w(j,k,N) = {if(k>N,1, 1/(1 - sum(i=j,N, x^i * w(i,k+1,N-i+1))))}
%o A384937 Ax(N) = {Vec(w(1,1,N)+ O('x^(N+1)))}
%o A384937 Ax(10)
%Y A384937 Cf. A000108, A000169, A000957, A001764, A036765, A384747, A384748.
%K A384937 nonn
%O A384937 0,3
%A A384937 _John Tyler Rascoe_, Jun 12 2025