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A301467
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Number of enriched r-trees of size n with no empty subtrees.
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25
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1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432
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OFFSET
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1,2
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COMMENTS
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An enriched r-tree of size n > 0 with no empty subtrees is either a single node of size n, or a finite nonempty sequence of enriched r-trees with no empty subtrees and with weakly decreasing sizes summing to n - 1.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1910
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FORMULA
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O.g.f.: x^2/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).
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EXAMPLE
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The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
5,
(4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
(31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
(211), ((1)11),
(1111).
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30); # Alois P. Heinz, Jun 21 2018
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MATHEMATICA
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pert[n_]:=pert[n]=If[n===1, 1, 1+Sum[Times@@pert/@y, {y, IntegerPartitions[n-1]}]];
Array[pert, 30]
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PROG
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(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); v} \\ Andrew Howroyd, Aug 26 2018
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CROSSREFS
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Cf. A000081, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A300660, A301342-A301345, A301364-A301368, A301422, A301462, A301469, A301470.
Sequence in context: A056952 A225585 A121703 * A275070 A115219 A078160
Adjacent sequences: A301464 A301465 A301466 * A301468 A301469 A301470
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Mar 21 2018
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STATUS
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approved
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