1,3

Number of distinct values taken by 0th and 1st derivative is 1,1,1,1,1,1,... and by 2nd is 1,1,2,3,4,5,...

Conjectured g.f.: (x^6-2*x^4-x^2+x-1)*x/(x^5-2*x^4+x^3-x^2+2*x-1). - Alois P. Heinz, Nov 02 2011

For n=5 there are 7 distinct values: 9, 15, 18, 21, 24, 33, 48, they are given by 3rd derivatives of the following parenthesizations at x=1: x^(x^((x^x)^x)), x^((x^(x^x))^x), (x^x)^(x^(x^x)), x^(((x^x)^x)^x), (x^(x^x))^(x^x), (((x^(x^x)))^x)^x, (((x^x)^x)^x)^x. So a(5)=7.

f:= proc(n) option remember;

`if`(n=1, {[0, 0]}, {seq(seq(seq([2+g[1], 3*(1+g[1]+h[1])+g[2]],

h=f(n-j)), g=f(j)), j=1..n-1)})

end:

a:= n-> nops(map(x-> x[2], f(n))):

seq(a(n), n=1..40); # Alois P. Heinz, Nov 03 2011

f[1] = {x}; f[n_] := Flatten[Table[Outer[Power, f[k], f[n-k]], {k, n-1}]]; Table[Length[Union[D[f[n], {x, 3}] /. x -> 1]], {n, 1, 8}] (* Reshetnikov *)

Table[If[n<3, 1, Floor[(n^2-2)/3]], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

Cf. A000081 (upper bound), A000108, A199205 (4th derivatives), A199296 (5th derivatives), A215703, A215842. Column k=3 of A216368.

Sequence in context: A347763 A198759 A078617 * A247184 A025703 A025709

Adjacent sequences: A199082 A199083 A199084 * A199086 A199087 A199088

Vladimir Reshetnikov, Nov 02 2011

a(13)-a(56) from Alois P. Heinz, Nov 02 2011

approved