A382096 Number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2,3}, and no nodes have the same weight as their parent node.

1, 1, 2, 6, 15, 39, 110, 308, 869, 2499, 7238, 21086, 61871, 182523, 540830, 1609238, 4805871, 14398559, 43264896, 130347450, 393650751, 1191441349, 3613345360, 10978726634, 33414836743, 101863289331, 310984519412, 950734751040, 2910319385881, 8919643999157, 27368321239074

Offset: 0

Author

John Tyler Rascoe, Jun 08 2025

Keywords

• nonn

Examples

			a(3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)
		

Crossrefs

Cf. A000108, A002212, A143330, A384613, A384685, (column k=3 of A384747).

Programs

PARI

b(i,j,k,N) = {if(k>N,1, 1/(1-sum(u=1,j, if(u==i,0,x^u*b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1-sum(i=1,k, b(i,k,1,N)*x^i)))}
Gx(3,20)

Formula

G.f.: G(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2 - b_3(x)*x^3) where b_1(x) = 1/(1 - b_2(x)*x^2 - b_3(x)*x^3), b_2(x) = 1/(1 - b_1(x)*x - b_3(x)*x^3), b_3(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2).