This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A358575 #7 Nov 25 2022 07:22:53
%S A358575 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 A358575 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 A358575 9,52,185,416,563,429,161,25,1
%N 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.
%e A358575 Triangle begins:
%e A358575 1
%e A358575 0 1
%e A358575 0 1 1
%e A358575 0 1 2 1
%e A358575 0 1 3 4 1
%e A358575 0 1 4 8 6 1
%e A358575 0 1 5 14 18 9 1
%e A358575 0 1 6 21 39 35 12 1
%e A358575 0 1 7 30 72 97 62 16 1
%e A358575 0 1 8 40 120 214 212 103 20 1
%e A358575 0 1 9 52 185 416 563 429 161 25 1
%t A358575 art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A358575 Table[Length[Select[art[n],Count[#,_[__],{0,Infinity}]==k&]],{n,1,10},{k,0,n-1}]
%Y A358575 Row sums are A000081.
%Y A358575 Column k = n - 2 appears to be A002620.
%Y A358575 Column k = 3 appears to be A006578.
%Y A358575 The version for height instead of internal nodes is A034781.
%Y A358575 Equals A055277 with rows reversed.
%Y A358575 The ordered version is A090181 or A001263.
%Y A358575 The central column is A185650, ordered A000891.
%Y A358575 The left half sums to A358583, strict A358581.
%Y A358575 The right half sums to A358584, strict A358582.
%Y A358575 Cf. A000108, A065097, A109129, A342507, A358578, A358587, A358589.
%K A358575 nonn,tabl
%O A358575 1,9
%A A358575 _Gus Wiseman_, Nov 23 2022