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A301364
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Regular triangle where T(n,k) is the number of enriched p-trees of weight n with k leaves.
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10
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1, 1, 1, 1, 1, 2, 1, 2, 4, 5, 1, 2, 6, 11, 12, 1, 3, 10, 26, 38, 34, 1, 3, 13, 39, 87, 117, 92, 1, 4, 19, 69, 181, 339, 406, 277, 1, 4, 23, 95, 303, 707, 1198, 1311, 806, 1, 5, 30, 143, 514, 1430, 2970, 4525, 4522, 2500, 1, 5, 35, 184, 762, 2446, 6124, 11627
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OFFSET
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1,6
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COMMENTS
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An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more enriched p-trees with weakly decreasing weights summing to n.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1
1 1 2
1 2 4 5
1 2 6 11 12
1 3 10 26 38 34
1 3 13 39 87 117 92
1 4 19 69 181 339 406 277
...
The T(5,4) = 11 enriched p-trees: (((21)1)1), ((2(11))1), (((11)2)1), ((211)1), ((21)(11)), (((11)1)2), ((111)2), ((21)11), (2(11)1), ((11)21), (2111).
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MATHEMATICA
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eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@ptn], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[Select[eptrees[n], Count[#, _Integer, {-1}]===k&]], {n, 8}, {k, n}]
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PROG
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(PARI) A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); apply(p->Vecrev(p/y), v)}
{ 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. A008284, A055277, A063834, A220418, A273866, A273873, A281145, A290261, A299201, A299202, A299203, A300354, A300442, A300443, A301365-A301368.
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KEYWORD
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AUTHOR
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STATUS
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approved
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