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A301342
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Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.
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13
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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, 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, 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
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OFFSET
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1,12
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LINKS
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EXAMPLE
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Triangle begins:
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 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 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
The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).
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MATHEMATICA
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irut[n_]:=irut[n]=If[n===1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[irut/@c]], UnsameQ@@#&]]/@IntegerPartitions[n-1]];
Table[Length[Select[irut[n], Count[#, {}, {-2}]===k&]], {n, 8}, {k, n}]
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CROSSREFS
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A version with the zeroes removed is A055327.
Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.
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KEYWORD
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AUTHOR
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STATUS
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approved
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