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A214570
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a(n) = Max(c(t)), where c(t) is the number of ordered trees isomorphic - as rooted trees - to the rooted tree t and the maximum is taken over all rooted trees with n vertices.
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3
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1, 1, 1, 2, 3, 4, 6, 12, 24, 40, 60, 120, 240, 420, 840, 1680
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OFFSET
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1,4
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COMMENTS
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a(n) is also the size of the largest equivalence class of function representations as x^x^...^x with n x's and parentheses inserted in all possible ways. a(4) = 2: (x^x)^(x^x) == (x^(x^x))^x; a(5) = 3: ((x^x)^x)^(x^x) == ((x^x)^(x^x))^x == ((x^(x^x))^x)^x. - Alois P. Heinz, Aug 31 2012
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LINKS
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FORMULA
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No formula available, except a(n)=number of entries in row n of A214569.
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EXAMPLE
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a(4) = 2 because among the four rooted trees with 4 vertices the path tree P_4, the star tree K_{1,3}, and the tree in the shape of Y are isomorphic only to themselves, while A - B - C - D with root at B is isomorphic to itself and to A - B - C - D with root at C.
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MAPLE
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F:= proc(n) option remember; `if`(n=1, [x+1],
[seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])
end:
a:= proc(n) option remember; local i, l, m, p; m:=0;
l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):
p:= proc() 0 end: forget(p);
for i in l do p(i):= p(i)+1; m:= max(m, p(i))
od: m
end:
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MATHEMATICA
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F[n_] := F[n] = If[n == 1, {x+1}, Flatten[Table[Table[Table[f^g, {g, F[n-i]}], {f, F[i]}], {i, 1, n-1}]]]; a[n_] := a[n] = Module[{i, l, m, p}, m = 0; l = Map[ Function[ {f}, Coefficient[Series[f, {x, 0, n+1}], x, n]], F[n]]; Clear[p]; p[_] = 0; Do[p[i] = p[i]+1; m = Max[m, p[i]], {i, l}]; m]; Table[a[n], {n, 1, 10}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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