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 A301344 #6 Mar 19 2018 22:06:25
%S A301344 1,1,0,1,1,0,1,2,0,0,1,4,1,0,0,1,6,4,0,0,0,1,9,11,2,0,0,0,1,12,24,9,0,
%T A301344 0,0,0,1,16,46,32,3,0,0,0,0,1,20,80,86,20,0,0,0,0,0,1,25,130,203,86,6,
%U A301344 0,0,0,0,0,1,30,200,423,283,46,0,0,0,0,0,0,1,36,295,816,786,234,11,0,0,0,0
%N A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.
%C A301344 A rooted tree is semi-binary if all outdegrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.
%e A301344 Triangle begins:
%e A301344 1
%e A301344 1 0
%e A301344 1 1 0
%e A301344 1 2 0 0
%e A301344 1 4 1 0 0
%e A301344 1 6 4 0 0 0
%e A301344 1 9 11 2 0 0 0
%e A301344 1 12 24 9 0 0 0 0
%e A301344 1 16 46 32 3 0 0 0 0
%e A301344 1 20 80 86 20 0 0 0 0 0
%e A301344 1 25 130 203 86 6 0 0 0 0 0
%e A301344 The T(6,3) = 4 semi-binary rooted trees: ((o(oo))), (o((oo))), (o(o(o))), ((o)(oo)).
%t A301344 rbt[n_]:=rbt[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rbt/@c]]]/@Select[IntegerPartitions[n-1],Length[#]<=2&]];
%t A301344 Table[Length[Select[rbt[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]
%Y A301344 Cf. A000081, A000598, A001190, A001678, A003238, A004111, A055277, A111299, A273873, A290689, A291636, A292050, A298118, A298204, A298422, A298426, A301342, A301343, A301345.
%K A301344 nonn,tabl
%O A301344 1,8
%A A301344 _Gus Wiseman_, Mar 19 2018