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.

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, 1560, 1800, 720, 0, 429, 1638, 3318, 8400, 16800, 15120, 5040, 0, 1430, 7278, 18180, 43128, 126000, 191520, 141120, 40320, 0, 4862, 32946, 98502, 238320, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

Author

John Tyler Rascoe, Jun 18 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    k=0    1    2      3     4      5      6     7
 n=0 [1]
 n=1 [0,   1]
 n=2 [0,   2,   2]
 n=3 [0,   5,   6,     6]
 n=4 [0,  14,  22,    36,   24]
 n=5 [0,  42,  90,   150,  240,   120]
 n=6 [0, 132,  378,  648, 1560,  1800,   720]
 n=7 [0, 429, 1638, 3318, 8400, 16800, 15120, 5040]
...
T(3,2) = 6 counts the three leaf permutations of each of the following trees:
      __o__        __o__
     /  |  \      /  |  \
   (1) (1) (2)  (1) (2) (2)
		

Crossrefs

Cf. A000108 (column k=1), A000142 (main diagonal), A385125 (row sums).

Cf. A107429, A384685, A384747.

Programs

PARI

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)}
C_aB(B) = {my(S = subsets(B)); sum(i=1,#S, (1/(1-x*z*#S[i]))*(-1)^(#B-#S[i]))}
D(k,N,B) = {if(k>N,1, substpol(C_aB(B),z,1 + D(k+1,N-#B+1,B)))}
Dx(N,B) = {Vec(1+D(1,N,B)+ O('x^(N+1)))}
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]))}
T(8)