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%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
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