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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.

%I A382096 #55 Jun 11 2025 03:59:32
%S A382096 1,1,2,6,15,39,110,308,869,2499,7238,21086,61871,182523,540830,
%T A382096 1609238,4805871,14398559,43264896,130347450,393650751,1191441349,
%U A382096 3613345360,10978726634,33414836743,101863289331,310984519412,950734751040,2910319385881,8919643999157,27368321239074
%N 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.
%F A382096 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).
%e A382096 a(3) = 6 counts:
%e A382096   o    o    o      o        o        __o__
%e A382096   |    |    |     / \      / \      /  |  \
%e A382096  (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
%e A382096        |    |
%e A382096       (1)  (2)
%o A382096 (PARI)
%o A382096 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)))))}
%o A382096 Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1-sum(i=1,k, b(i,k,1,N)*x^i)))}
%o A382096 Gx(3,20)
%Y A382096 Cf. A000108, A002212, A143330, A384613, A384685, (column k=3 of A384747).
%K A382096 nonn
%O A382096 0,3
%A A382096 _John Tyler Rascoe_, Jun 08 2025