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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

%I #7 Nov 25 2022 07:22:53

%S 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,

%T 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,

%U 9,52,185,416,563,429,161,25,1

%N 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.

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 1 2 1

%e 0 1 3 4 1

%e 0 1 4 8 6 1

%e 0 1 5 14 18 9 1

%e 0 1 6 21 39 35 12 1

%e 0 1 7 30 72 97 62 16 1

%e 0 1 8 40 120 214 212 103 20 1

%e 0 1 9 52 185 416 563 429 161 25 1

%t art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];

%t Table[Length[Select[art[n],Count[#,_[__],{0,Infinity}]==k&]],{n,1,10},{k,0,n-1}]

%Y Row sums are A000081.

%Y Column k = n - 2 appears to be A002620.

%Y Column k = 3 appears to be A006578.

%Y The version for height instead of internal nodes is A034781.

%Y Equals A055277 with rows reversed.

%Y The ordered version is A090181 or A001263.

%Y The central column is A185650, ordered A000891.

%Y The left half sums to A358583, strict A358581.

%Y The right half sums to A358584, strict A358582.

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

%K nonn,tabl

%O 1,9

%A _Gus Wiseman_, Nov 23 2022