

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.


10



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
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OFFSET

1,3


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,...


LINKS

Table of n, a(n) for n=1..56.


FORMULA

Conjectured g.f.: (x^62*x^4x^2+x1)*x/(x^52*x^4+x^3x^2+2*x1).  Alois P. Heinz, Nov 02 2011


EXAMPLE

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.


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(nj)), g=f(j)), j=1..n1)})
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[nk]], {k, n1}]]; Table[Length[Union[D[f[n], {x, 3}] /. x > 1]], {n, 1, 8}] (* Reshetnikov *)
Table[If[n<3, 1, Floor[(n^22)/3]], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)


CROSSREFS

Cf. A000081 (upper bound), A000108, A199205 (4th derivatives), A199296 (5th derivatives), A215703, A215842. Column k=3 of A216368.
Sequence in context: A194168 A198759 A078617 * A247184 A025703 A025709
Adjacent sequences: A199082 A199083 A199084 * A199086 A199087 A199088


KEYWORD

nonn


AUTHOR

Vladimir Reshetnikov, Nov 02 2011


EXTENSIONS

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


STATUS

approved



