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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.
15
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
1,9
EXAMPLE
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
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}]
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.
Sequence in context: A353436 A286932 A350364 * A259475 A361952 A323224
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 23 2022
STATUS
approved