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A301422
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Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.
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22
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 19, 14, 5, 1, 0, 1, 12, 36, 40, 21, 6, 1, 0, 1, 16, 65, 102, 75, 30, 7, 1, 0, 1, 20, 106, 223, 224, 123, 40, 8, 1, 0, 1, 25, 168, 457, 604, 439, 191, 52, 9, 1, 0, 1, 30, 248, 847, 1433, 1346, 764, 276
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OFFSET
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1,8
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COMMENTS
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An r-tree (A093637) of size n > 0 is a finite sequence of r-trees with weakly decreasing sizes summing to n - 1. This is a similar construction to p-trees (A196545) except that r-trees are not required to be series-reduced and are weighted by all nodes (including the root) rather than just the leaves.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
1 1 0
1 2 1 0
1 4 3 1 0
1 6 8 4 1 0
1 9 19 14 5 1 0
1 12 36 40 21 6 1 0
1 16 65 102 75 30 7 1 0
1 20 106 223 224 123 40 8 1 0
1 25 168 457 604 439 191 52 9 1 0
...
The T(6,3) = 8 r-trees: (((ooo))), (((oo)o)), (((o)oo)), (((oo))o), (((o)o)o), ((oo)(o)), (((o))oo), ((o)(o)o).
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MATHEMATICA
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rtrees[n_]:=Join@@Table[Tuples[rtrees/@y], {y, IntegerPartitions[n-1]}];
Table[Length[Select[rtrees[n], Count[#, {}, {-2}]===k&]], {n, 8}, {k, n}]
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PROG
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(PARI) A(n)={my(v=vector(n)); v[1]=y; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); vector(n, k, Vecrev(v[k]/y, k))}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018
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CROSSREFS
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Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A291443, A297791, A300443, A301342-A301345, A301364.
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KEYWORD
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AUTHOR
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STATUS
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approved
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