1,4

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

No formula available, except a(n)=number of entries in row n of A214569.

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.

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:

seq(a(n), n=1..10); # Alois P. Heinz, Aug 31 2012

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 *)

Cf. A214569, A206487, A214571, A215703.

Sequence in context: A277408 A018369 A324178 * A306348 A363670 A078495

Adjacent sequences: A214567 A214568 A214569 * A214571 A214572 A214573

nonn,hard,more

Emeric Deutsch, Jul 28 2012

a(12)-a(16) from Alois P. Heinz, Sep 06 2012

approved