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A384685 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, all internal nodes have weight 1, and leaf nodes have weights in {1,...,k}.

%I A384685 #8 Jun 07 2025 08:22:57
%S A384685 1,0,1,0,2,3,0,5,8,9,0,14,25,28,29,0,42,83,95,98,99,0,132,289,337,349,
%T A384685 352,353,0,429,1041,1236,1285,1297,1300,1301,0,1430,3847,4652,4854,
%U A384685 4903,4915,4918,4919,0,4862,14504,17865,18709,18912,18961,18973,18976,18977
%N A384685 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, all internal nodes have weight 1, and leaf nodes have weights in {1,...,k}.
%F A384685 G.f. of column k is (1 - b(k,x) - sqrt((b(k,x) - 1)^2 - 4*x))/(2*x) where b(k,x) = (x^2 - x^(k + 1))/(1 - x).
%F A384685 T(n,k) = T(n,n) for k > n.
%e A384685 Triangle begins:
%e A384685     k=0     1     2     3     4     5     6     7      8
%e A384685  n=0 [1]
%e A384685  n=1 [0,    1]
%e A384685  n=2 [0,    2,    3]
%e A384685  n=3 [0,    5,    8,    9]
%e A384685  n=4 [0,   14,   25,   28,   29]
%e A384685  n=5 [0,   42,   83,   95,   98,   99]
%e A384685  n=6 [0,  132,  289,  337,  349,  352,  353]
%e A384685  n=7 [0,  429, 1041, 1236, 1285, 1297, 1300, 1301]
%e A384685  n=8 [0, 1430, 3847, 4652, 4854, 4903, 4915, 4918, 4919]
%e A384685 ...
%e A384685 T(2,2) = 3 counts:
%e A384685   o    o      o
%e A384685   |    |     / \
%e A384685  (2)  (1)  (1) (1)
%e A384685        |
%e A384685       (1)
%o A384685 (PARI)
%o A384685 b(k) = {(x^2-x^(k+1))/(1-x)}
%o A384685 P(N,k) = {my(x='x+O('x^N)); Vec((1-b(k)-sqrt((b(k)-1)^2-4*x))/(2*x))}
%o A384685 T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1,1,0))~); for(k=1,N, v=matconcat([v,P(N+1,k)~])); vector(N,n, vector(n,k,v[n,k]))}
%Y A384685 Cf. (column k=1) A000108, A078481, A078482, A088218, (column k=2) A143330, A380761, A384613.
%K A384685 nonn,easy,tabl
%O A384685 0,5
%A A384685 _John Tyler Rascoe_, Jun 06 2025