A358575 Triangle read by rows where T(n,k) is the number of unlabeled n-node rooted trees with k = 0..n-1 internal (non-leaf) nodes.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 6, 1, 0, 1, 5, 14, 18, 9, 1, 0, 1, 6, 21, 39, 35, 12, 1, 0, 1, 7, 30, 72, 97, 62, 16, 1, 0, 1, 8, 40, 120, 214, 212, 103, 20, 1, 0, 1, 9, 52, 185, 416, 563, 429, 161, 25, 1

Offset: 1

Author

Gus Wiseman, Nov 23 2022

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1
    0    1
    0    1    1
    0    1    2    1
    0    1    3    4    1
    0    1    4    8    6    1
    0    1    5   14   18    9    1
    0    1    6   21   39   35   12    1
    0    1    7   30   72   97   62   16    1
    0    1    8   40  120  214  212  103   20    1
    0    1    9   52  185  416  563  429  161   25    1
		

Crossrefs

Row sums are A000081.

Column k = n - 2 appears to be A002620.

Column k = 3 appears to be A006578.

The version for height instead of internal nodes is A034781.

Equals A055277 with rows reversed.

The ordered version is A090181 or A001263.

The central column is A185650, ordered A000891.

The left half sums to A358583, strict A358581.

The right half sums to A358584, strict A358582.

Cf. A000108, A065097, A109129, A342507, A358578, A358587, A358589.

Programs

Mathematica

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,[_],{0,Infinity}]==k&]],{n,1,10},{k,0,n-1}]