A301345 - OEIS

%I #5 Mar 19 2018 22:06:31

%S 1,1,0,0,1,0,0,1,1,0,0,0,2,1,0,0,0,1,3,1,0,0,0,1,2,4,1,0,0,0,0,3,4,5,

%T 1,0,0,0,0,2,6,6,6,1,0,0,0,0,1,6,10,9,7,1,0,0,0,0,1,5,12,16,12,8,1,0,

%U 0,0,0,0,4,13,22,23,16,9,1,0,0,0,0,0,3,14,27,36,32,20,10,1,0,0,0,0,0,2,11

%N Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

%e Triangle begins:

%e 1

%e 1   0

%e 0   1   0

%e 0   1   1   0

%e 0   0   2   1   0

%e 0   0   1   3   1   0

%e 0   0   1   2   4   1   0

%e 0   0   0   3   4   5   1   0

%e 0   0   0   2   6   6   6   1   0

%e 0   0   0   1   6  10   9   7   1   0

%e 0   0   0   1   5  12  16  12   8   1   0

%e The T(9,5) = 6 transitive rooted trees: (o(o)(oo(o))), (o((oo))(oo)), (oo(o)(o(o))), (o(o)(o)(oo)), (ooo(o)((o))), (oo(o)(o)(o)).

%t rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];

%t trat[n_]:=Select[rut[n],Complement[Union@@#,#]==={}&];

%t Table[Length[Select[trat[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

%Y Row sums are A290689.

%Y Cf. A000081, A001190, A003238, A004111, A032305, A055277, A276625, A279861, A290760, A290822, A298422, A298426, A301342, A301343, A301344.

%K nonn,tabl

%O 1,13

%A _Gus Wiseman_, Mar 19 2018