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

0,3

Links

Table of n, a(n) for n=0..19.

Formula

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

Example

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

Prog

(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)))}

Crossrefs

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

Sequence in context: A277610 A261899 A109186 * A099391 A008785 A375616

Adjacent sequences: A384610 A384611 A384612 * A384614 A384615 A384616

Keyword

nonn

Author

John Tyler Rascoe, Jun 04 2025

Status

approved