A214570 - OEIS

%I #29 May 28 2015 10:30:39

%S 1,1,1,2,3,4,6,12,24,40,60,120,240,420,840,1680

%N 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.

%C 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

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

%e 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.

%p F:= proc(n) option remember; `if`(n=1, [x+1],

%p       [seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])

%p     end:

%p a:= proc(n) option remember; local i, l, m, p; m:=0;

%p       l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):

%p       p:= proc() 0 end: forget(p);

%p       for i in l do p(i):= p(i)+1; m:= max(m, p(i))

%p       od: m

%p     end:

%p seq(a(n), n=1..10);  # _Alois P. Heinz_, Aug 31 2012

%t 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_ *)

%Y Cf. A214569, A206487, A214571, A215703.

%K nonn,hard,more

%O 1,4

%A _Emeric Deutsch_, Jul 28 2012

%E a(12)-a(16) from _Alois P. Heinz_, Sep 06 2012