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.

1, 1, 5, 36, 340, 4019, 57696, 982146, 19419042, 438068191, 11106513798, 312555754796, 9663786464541, 325515760762637, 11861723942987878, 464834173383876612, 19490387161582849600, 870582781070074780946, 41266849779858887379029, 2068827708558025551348644

Offset: 0

Author

John Tyler Rascoe, Jun 04 2025

Keywords

• nonn

Examples

			a(2) = 5 counts:
   o      o        o        o        o
   |     / \      / \      / \      / \
   o   (1) (1)  (1) (2)  (2) (1)  (2) (2)
   |
  (1)
		

Crossrefs

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

Programs

PARI

G(k,N) = if(k>N, 1, 1+ sum(i=1,N, (x*(G(k+1,N-i+1)+i-1))^i))
G_x(N) = {my(x='x+O('x^N)); Vec(G(1,N)+ O('x^(N+1)))}

Formula

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