A199085 Number of distinct values taken by 3rd derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

1, 1, 2, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 64, 74, 84, 95, 107, 119, 132, 146, 160, 175, 191, 207, 224, 242, 260, 279, 299, 319, 340, 362, 384, 407, 431, 455, 480, 506, 532, 559, 587, 615, 644, 674, 704, 735, 767, 799, 832, 866, 900, 935, 971, 1007, 1044

Offset: 1

Author

Vladimir Reshetnikov, Nov 02 2011

Keywords

• nonn

Comments

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,...

Examples

			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.
		

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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