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A108521
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Number of rooted trees with n generators.
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11
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1, 2, 5, 16, 53, 194, 730, 2868, 11526, 47370, 197786, 837467, 3585696, 15501423, 67563442, 296579626, 1309973823, 5817855174, 25964218471, 116379947718, 523699384013, 2364967753113, 10714396241046, 48684193997623
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OFFSET
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1,2
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COMMENTS
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A generator is a leaf or a node with just one child.
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LINKS
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FORMULA
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G.f.: satisfies (2-x)*A(x) = x - 1 + EULER(A(x)).
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = 1+a[n-1]+Total[Product[Binomial[a[i]-1+Count[#, i], Count[#, i]], {i, DeleteCases[DeleteDuplicates[#], 1]}]&/@ IntegerPartitions[n, {2, n-1}]]; Table[a[n], {n, 24}] (* Robert A. Russell, Jun 02 2020 *)
a[1] = 1; a[n_] := a[n] = a[n-1] + (DivisorSum[n, a[#] # &, #<n &] + Sum[c[k] b[n-k], {k, 1, n-1}])/n; b[n_] := b[n] = (c[n] + Sum[c[k] b[n-k], {k, 1, n-1}])/n; c[n_] := c[n] = DivisorSum[n, a[#] # &]; Table[a[k], {k, 24}] (* Robert A. Russell, Jun 04 2020 *)
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat(v, v[#v] + EulerT(concat(v, [0]))[n])); v} \\ Andrew Howroyd, Aug 31 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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