cp's OEIS Frontend

A384938 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 strictly increasing.

%I A384938 #7 Jun 14 2025 00:18:19
%S A384938 1,1,2,5,11,26,61,142,334,785,1845,4339,10211,24030,56560,133143,
%T A384938 313433,737906,1737275,4090206,9630067,22673482,53383917,125691264,
%U A384938 295938451,696785116,1640579144,3862745470,9094847357,21413863699,50419073794,118712060012,279508439419
%N A384938 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 strictly increasing.
%F A384938 G.f.: G_0(x) where G_k(x) = 1/(1 - Sum_{i>k} x^i * G_i(x)).
%e A384938 The following tree with sum of edge weights 15 contains downward paths of edge weights (1), (2,3,4), and (2,3,5) all of which are weakly increasing. So this tree is counted under a(13) = 133143.
%e A384938            o
%e A384938         2 / \ 1
%e A384938          o   o
%e A384938       3 / 	
%e A384938        o
%e A384938     4 / \ 5	
%e A384938      o   o
%o A384938 (PARI)
%o A384938 w(j,k,N) = {if(k>N,1, 1/(1 - sum(i=j+1,N, x^i * w(i,k+1,N-i+1))))}
%o A384938 Bx(N) = {my(x='x+O('x^(N+1))); Vec(w(0,1,N)+ O('x^(N+1)))}
%o A384938 Bx(10)
%Y A384938 Cf. A000108, A000169, A000957, A001764, A036765, A384747, A384748.
%K A384938 nonn
%O A384938 0,3
%A A384938 _John Tyler Rascoe_, Jun 13 2025