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A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

%I A384747 #16 Jun 12 2025 00:50:46
%S A384747 1,0,1,0,1,2,0,1,5,6,0,1,11,15,16,0,1,26,39,43,44,0,1,63,110,123,127,
%T A384747 128,0,1,153,308,358,371,375,376,0,1,376,869,1046,1096,1109,1113,1114,
%U A384747 0,1,931,2499,3098,3278,3328,3341,3345,3346,0,1,2317,7238,9283,9904,10084,10134,10147,10151,10152
%N A384747 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.
%F A384747 T(n,k) = T(n,n) for k > n.
%e A384747 Triangle begins:
%e A384747     k=0  1    2     3     4     5     6     7     8     9
%e A384747  n=0 [1]
%e A384747  n=1 [0, 1]
%e A384747  n=2 [0, 1,   2]
%e A384747  n=3 [0, 1,   5,    6]
%e A384747  n=4 [0, 1,  11,   15,   16]
%e A384747  n=5 [0, 1,  26,   39,   43,   44]
%e A384747  n=6 [0, 1,  63,  110,  123,  127,  128]
%e A384747  n=7 [0, 1, 153,  308,  358,  371,  375,  376]
%e A384747  n=8 [0, 1, 376,  869, 1046, 1096, 1109, 1113, 1114]
%e A384747  n=9 [0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346]
%e A384747 ...
%e A384747 T(3,3) = 6 counts:
%e A384747   o    o    o      o        o        __o__
%e A384747   |    |    |     / \      / \      /  |  \
%e A384747  (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
%e A384747        |    |
%e A384747       (1)  (2)
%o A384747 (PARI)
%o A384747 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 A384747 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 A384747 T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Gx(k,N)~])); vector(N, n, vector(n, k, v[n, k]))}
%o A384747 T(9)
%Y A384747 Cf. A051286 (column k=2), A382096 (column k=3), A384748 (main diagonal).
%Y A384747 Cf. A000108, A002212, A143330, A384613, A384685.
%K A384747 nonn,tabl
%O A384747 0,6
%A A384747 _John Tyler Rascoe_, Jun 09 2025