cp's OEIS Frontend

A384613 Number of rooted ordered trees with n non-root nodes such that all leaf nodes can be k different colors where k is the degree of their parent node.

%I A384613 #6 Jun 05 2025 09:56:05
%S A384613 1,1,5,36,340,4019,57696,982146,19419042,438068191,11106513798,
%T A384613 312555754796,9663786464541,325515760762637,11861723942987878,
%U A384613 464834173383876612,19490387161582849600,870582781070074780946,41266849779858887379029,2068827708558025551348644
%N A384613 Number of rooted ordered trees with n non-root nodes such that all leaf nodes can be k different colors where k is the degree of their parent node.
%F A384613 G.f.: G(x) satisfies G(x) = Sum_{i>=0} (x*(G(x) + i - 1))^i.
%e A384613 a(2) = 5 counts:
%e A384613    o      o        o        o        o
%e A384613    |     / \      / \      / \      / \
%e A384613    o   (1) (1)  (1) (2)  (2) (1)  (2) (2)
%e A384613    |
%e A384613   (1)
%o A384613 (PARI)
%o A384613 G(k,N) = if(k>N, 1, 1+ sum(i=1,N, (x*(G(k+1,N-i+1)+i-1))^i))
%o A384613 G_x(N) = {my(x='x+O('x^N)); Vec(G(1,N)+ O('x^(N+1)))}
%Y A384613 Cf. A000108, A002212, A007318, A102407, A258173.
%K A384613 nonn
%O A384613 0,3
%A A384613 _John Tyler Rascoe_, Jun 04 2025