A301342 - OEIS

%I #5 Mar 19 2018 22:06:07

%S 1,1,0,1,0,0,1,1,0,0,1,2,0,0,0,1,4,1,0,0,0,1,6,5,0,0,0,0,1,9,13,2,0,0,

%T 0,0,1,12,28,11,0,0,0,0,0,1,16,53,40,3,0,0,0,0,0,1,20,91,109,26,0,0,0,

%U 0,0,0,1,25,146,254,116,6,0,0,0,0,0,0,1,30,223,524,387,61,0,0,0,0,0,0,0,1,36

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

%e Triangle begins:

%e 1

%e 1   0

%e 1   0   0

%e 1   1   0   0

%e 1   2   0   0   0

%e 1   4   1   0   0   0

%e 1   6   5   0   0   0   0

%e 1   9  13   2   0   0   0   0

%e 1  12  28  11   0   0   0   0   0

%e 1  16  53  40   3   0   0   0   0   0

%e 1  20  91 109  26   0   0   0   0   0   0

%e 1  25 146 254 116   6   0   0   0   0   0   0

%e 1  30 223 524 387  61   0   0   0   0   0   0   0

%e The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).

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

%t Table[Length[Select[irut[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]

%Y A version with the zeroes removed is A055327.

%Y Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.

%K nonn,tabl

%O 1,12

%A _Gus Wiseman_, Mar 19 2018