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A385123 Triangle Read by rows: T(n,k) is the number of rooted ordered trees with n non-root nodes with non-root node labels in {1,..,k} such that all labels appear at least once in all groups of sibling nodes.

%I A385123 #7 Jun 22 2025 03:11:35
%S A385123 1,0,1,0,2,2,0,5,6,6,0,14,22,36,24,0,42,90,150,240,120,0,132,378,648,
%T A385123 1560,1800,720,0,429,1638,3318,8400,16800,15120,5040,0,1430,7278,
%U A385123 18180,43128,126000,191520,141120,40320,0,4862,32946,98502,238320,834120,1905120,2328480,1451520,362880
%N A385123 Triangle Read by rows: T(n,k) is the number of rooted ordered trees with n non-root nodes with non-root node labels in {1,..,k} such that all labels appear at least once in all groups of sibling nodes.
%e A385123 Triangle begins:
%e A385123     k=0    1    2      3     4      5      6     7
%e A385123  n=0 [1]
%e A385123  n=1 [0,   1]
%e A385123  n=2 [0,   2,   2]
%e A385123  n=3 [0,   5,   6,     6]
%e A385123  n=4 [0,  14,  22,    36,   24]
%e A385123  n=5 [0,  42,  90,   150,  240,   120]
%e A385123  n=6 [0, 132,  378,  648, 1560,  1800,   720]
%e A385123  n=7 [0, 429, 1638, 3318, 8400, 16800, 15120, 5040]
%e A385123 ...
%e A385123 T(3,2) = 6 counts the three leaf permutations of each of the following trees:
%e A385123       __o__        __o__
%e A385123      /  |  \      /  |  \
%e A385123    (1) (1) (2)  (1) (2) (2)
%o A385123 (PARI)
%o A385123 subsets(S) = {my(s=List()); for(i=0, 2^(#S) -1, my(x=List()); for(j=1,#S, if(bitand(i, 1<<(j-1)), listput(x, S[j]))); listput(s,Vec(x))); Vec(s)}
%o A385123 C_aB(B) = {my(S = subsets(B)); sum(i=1,#S, (1/(1-x*z*#S[i]))*(-1)^(#B-#S[i]))}
%o A385123 D(k,N,B) = {if(k>N,1, substpol(C_aB(B),z,1 + D(k+1,N-#B+1,B)))}
%o A385123 Dx(N,B) = {Vec(1+D(1,N,B)+ O('x^(N+1)))}
%o A385123 T(max_row) = {my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Dx(N+1, vector(k,i,i))~])); vector(N, n, vector(n, k, v[n, k]))}
%o A385123 T(8)
%Y A385123 Cf. A000108 (column k=1), A000142 (main diagonal), A385125 (row sums).
%Y A385123 Cf. A107429, A384685, A384747.
%K A385123 nonn,tabl
%O A385123 0,5
%A A385123 _John Tyler Rascoe_, Jun 18 2025