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.
Triangle begins: 1 1 1 1 0 1 1 1 2 2 1 0 0 0 1 1 1 1 5 3 3 1 0 0 0 0 0 1 1 1 2 6 12 14 12 6 1 0 1 0 3 0 3 0 2 1 1 0 0 1 7 10 10 5 3 1 0 0 0 0 0 0 0 0 0 1 1 1 3 7 21 41 58 100 100 94 48 20 The T(8,4) = 6 same-trees: (4(2(11))), (4((11)2)), ((22)(22)), ((2(11))4), (((11)2)4), (2222).
sametrees[n_]:=Prepend[Join@@Table[Tuples[sametrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[sametrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]
A(n)={my(v=vector(n)); for(n=1, n, v[n] = x + sumdiv(n, d, v[n/d]^d)); apply(p -> Vecrev(p/x), v)}
{my(v=A(16)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Aug 20 2018
The a(1) = 0 through a(7) = 7 trees:
. (o) . ((oo)) ((o)(o)) (((ooo))) (((o))(oo))
(o(o)) ((o(oo))) (((o)(oo)))
((oo(o))) ((o)((oo)))
(o((oo))) ((o)(o(o)))
(o(o(o))) ((o(o)(o)))
(oo((o))) (o((o)(o)))
(o(o)((o)))
art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]
\\ Needs R(n,f) defined in A358589. seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023
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