cp's OEIS Frontend

A214570 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.

%I A214570 #29 May 28 2015 10:30:39
%S A214570 1,1,1,2,3,4,6,12,24,40,60,120,240,420,840,1680
%N A214570 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 A214570 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 A214570 No formula available, except a(n)=number of entries in row n of A214569.
%e A214570 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 A214570 F:= proc(n) option remember; `if`(n=1, [x+1],
%p A214570       [seq(seq(seq(f^g, g=F(n-i)), f=F(i)), i=1..n-1)])
%p A214570     end:
%p A214570 a:= proc(n) option remember; local i, l, m, p; m:=0;
%p A214570       l:= map(f->coeff(series(f, x, n+1), x, n), F(n)):
%p A214570       p:= proc() 0 end: forget(p);
%p A214570       for i in l do p(i):= p(i)+1; m:= max(m, p(i))
%p A214570       od: m
%p A214570     end:
%p A214570 seq(a(n), n=1..10);  # _Alois P. Heinz_, Aug 31 2012
%t A214570 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 A214570 Cf. A214569, A206487, A214571, A215703.
%K A214570 nonn,hard,more
%O A214570 1,4
%A A214570 _Emeric Deutsch_, Jul 28 2012
%E A214570 a(12)-a(16) from _Alois P. Heinz_, Sep 06 2012