A384613 - OEIS

%I #6 Jun 05 2025 09:56:05

%S 1,1,5,36,340,4019,57696,982146,19419042,438068191,11106513798,

%T 312555754796,9663786464541,325515760762637,11861723942987878,

%U 464834173383876612,19490387161582849600,870582781070074780946,41266849779858887379029,2068827708558025551348644

%N 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 G.f.: G(x) satisfies G(x) = Sum_{i>=0} (x*(G(x) + i - 1))^i.

%e a(2) = 5 counts:

%e    o      o        o        o        o

%e    |     / \      / \      / \      / \

%e    o   (1) (1)  (1) (2)  (2) (1)  (2) (2)

%e    |

%e   (1)

%o (PARI)

%o G(k,N) = if(k>N, 1, 1+ sum(i=1,N, (x*(G(k+1,N-i+1)+i-1))^i))

%o G_x(N) = {my(x='x+O('x^N)); Vec(G(1,N)+ O('x^(N+1)))}

%Y Cf. A000108, A002212, A007318, A102407, A258173.

%K nonn

%O 0,3

%A _John Tyler Rascoe_, Jun 04 2025